Personally, I'm with you. If I were doing the numbering, Pluto would be Sol IX. So apply the question to Ceres, Eris, and our other little balls of rock which were never granted planet status in the first place.
Ceres was considerd a planet when first discovered. The first four asteroids were considered planets when first discovered. Some astronomers even classified the asteroids down to number 15 as planets, until in the 1850s they wre reclassified as minor planets and asteroids.
https://en.wikipedia.org/wiki/List_of_former_planets
Wikipedia has a list of objects which were formerly considered to be planets for longer or shorter periods of times. So at different times there would have been different numbers of planets listed in astronomy books that you might think.
Perhaps I should give a brief, a very brief, discussion of the names of stars and extra solar planets in science and science fiction.
For countless thousands of years humans on Earth have been giving proper names to various stars, often based on their apparent positions in the constellations imagined in the stars. So any particularly bright star will have many different proper names.
A man named Bayer published a star atlas in 1603. The plates depicting each constellation gave lower case Greek letters to the naked eye visible stars in each constellation, usually, but not always, in order of declining apparent brightness. So astronomers got into the habit of designating stars by the their Greek letters (often spelled out in the Latin alphabet) and the Latin genitive case of the constellation name.
A century later Astronomer Royale John Flamsteed created a star catalog. The naked eye visible stars in each constellation were given numbers in order of their increasing right ascension coordinates, followed by the Latin genitive case of the constellation name.
And probably hundreds of star catalogs have been published since, some with hundreds of stars, some with hundreds of thousands of stars, and I think a few with hundreds of millions of stars, though most of the stars in our Milky Way Galaxy have no designation in any catalog.
A star is typically designated by the full name or standard abbreviation of a catalog, followed by its number in that catalog. Well known stars can have designations in dozens of different catalogs.
Thus the star that fans usually think that Vulcan orbits has a Bayer designation of Omicron 2 Eridani (the 2 is in superscript position) and a Flamsteed designation of 40 Eridani, and many other catalog numbers.
In this millenium the International Astronomical Union created a committee which has been given official proper names to naked eye stars, usually using names which have been used for centuries or millennia in some Earth cultures.
Many stars are double or multiple star systems. Individual stars in those systems are known by the system name or designation followed by A, B, C, D, etc. in order of luminosity. If a component of a system is later found to be itself two stars, their original designationis followed by a, or b in lower case.
The convention for exoplanets discovered orbiting other stars is usually to give them lower case Latin letters, starting with b, in order of their discovery. If astronomers gave the letters in order of distance from the star, they would often have to reletter planets after new planets closer tot he star were discovered.
40 Eridani is a triple star, consisting of 40 Eridani A, B, and C. 40 Eridani A now officially has the name of Keid, which was givenn to the entire system centuries before the three separate stars were discovered.
So the first planet discovered to orbit Omicron 2 Eridani.A, or 40 eriani A, or Keid, will be designated Omicron 2 Eridani.A b, or 40 eriani A b, or Keid b. At present i here is no confirmed planet orbiting that star.
The four big moons of Jupiter were discovered by Galileo in 1610 and by Simon Marius soon after. Simon Marius named them Io, Europa, Ganymede, and Callisto in increasing order of distance from Jupiter, but astronomers rarely used them
The Galilean moons of Jupiter (
Io,
Europa,
Ganymede, and
Callisto) were named by
Simon Marius soon after their discovery in 1610.
[26] However, these names fell out of favor until the 20th century. The astronomical literature instead simply referred to "Jupiter I", "Jupiter II", etc., or "the first satellite of Jupiter", "Jupiter's second satellite", and so on.
[26] The names Io, Europa, Ganymede, and Callisto became popular in the mid-20th century,
[27] whereas the rest of the moons remained unnamed and were usually numbered in Roman numerals V (5) to XII (12).
[28][29] Jupiter V was discovered in 1892 and given the name
Amalthea by a popular though unofficial convention, a name first used by French astronomer
Camille Flammarion.
[19][30]
The other moons were simply labeled by their Roman numeral (e.g. Jupiter IX) in the majority of astronomical literature until the 1970s.
[31] Several different suggestions were made for names of Jupiter's outer satellites, but none were universally accepted until 1975 when the
International Astronomical Union's (IAU) Task Group for Outer Solar System Nomenclature granted names to satellites V–XIII,
[32] and provided for a formal naming process for future satellites still to be discovered.
[32] The practice was to name newly discovered moons of Jupiter after lovers and favorites of the god
Jupiter (
Zeus) and, since 2004, also after their descendants.
[19] All of Jupiter's satellites from XXXIV (
Euporie) onward are named after descendants of Jupiter or Zeus,
[19] except LIII (
Dia), named after a lover of Jupiter. Names ending with "a" or "o" are used for prograde irregular satellites (the latter for highly inclined satellites), and names ending with "e" are used for retrograde irregulars.
[33] With the discovery of smaller, kilometre-sized moons around Jupiter, the IAU has established an additional convention to limit the naming of small moons with
absolute magnitudes greater than 18 or diameters smaller than 1 km (0.62 mi).
[34] Some of the most recently confirmed moons have not received names.
https://en.wikipedia.org/wiki/Moons_of_Jupiter
The Roman numbering system for satellites arose with the very first discovery of natural satellites other than Earth's Moon:
Galileo referred to the
Galilean moons as
I through
IV (counting from Jupiter outward), refusing to adopt the names proposed by his rival
Simon Marius. Similar numbering schemes naturally arose with the discovery of multiple moons around Saturn, Uranus, and Mars. The numbers initially designated the moons in orbital sequence, and were re-numbered after each new discovery; for instance, before the discovery of Mimas and Enceladus in 1789, Tethys was Saturn I, Dione Saturn II, etc.,
[28] but after the new moons were discovered, Mimas became Saturn I, Enceladus Saturn II, Tethys Saturn III and Dione Saturn IV.
n the middle of the 19th century, however, the numeration became fixed, and later discoveries failed to conform with the orbital sequence scheme.
Amalthea, discovered in 1892, was labelled "Jupiter V" although it orbits more closely to Jupiter than does
Io (Jupiter I). The unstated convention then became, at the close of the 19th century, that the numbers more or less reflected the order of discovery, except for prior historical exceptions (see
Timeline of discovery of Solar System planets and their natural satellites); though if a large number of satellites were discovered in a short span of time, the group could be numbered in orbital sequence, or according to other principles than strictly by order of discovery. The convention has been extended to natural satellites of minor planets, such as
(87) Sylvia I Romulus. The outer irregular satellites of Jupiter (VI through XII) were left officially unnamed throughout this period, although as stated above some unofficial names were used in some contexts.
From 1975 to 2009, the
International Astronomical Union was assigning names to all planetary satellites, and Roman numerals were usually not assigned to satellites until they are named. (An exception is Saturn's moon
Helene, which received the Roman numeral XII in 1982, but was not named until 1988.) During this period, the use of Roman numeral designations diminished, and some are very rarely used;
Phobos and
Deimos are rarely referred to as Mars I and Mars II, and the
Moon is never referred to as "Earth I". However, since 2015 some moons have again been numbered without being named, starting from
Jupiter LI.
https://en.wikipedia.org/wiki/Naming_of_moons#Roman_numeral_designations
As pulp science fiction developed various conventions in the first half of the 20th century, alien planets around different stars could be named many ways. They could have proper names given by their intlligent natives or Earth Earth explorers. And their stars could be known by native names, or names given by Earth explorers, or by real names used on Earth at the time the stories were written.
Science fiction developed a convention that planets could be known by the human or native name or catelog designation of the star, followed by a Roman numeral in order of increasing orbital distance from the star. So in our solar system Mercury would be Sol I, Venus Sol II, Earth Sol III, and so on.
For example, in E.E. Smith's
Second Stage Lensman the planet Tralle is also known as Trallis III, and the planet Onlo in the same star system is also kown as Trallis IX.
And in the early 20th century there were less than 31 moons known in the solar system, some of them with Roman numberals in orbital order and some of them in order of discovery.
Thus it is a historical question which science fiction writers originated the science fiction numbering convention.
Second Stage Lensman was orginally serialized in
astrounding Science Fiction NObember 1941 to February 1942, but I have only read the book version, first published in 1953, and I don't know if the planetary numbers were added when it was revised for the book.
Anyway,
Star Trek more or less follows the naming conventions of science fiction. Many stars have proper names, some from Earth, many stars have Bayer designations, many stars have designations in probably fictional catalogs, etc. And similarly many planets are known by proper names, and others are known by the name or designatinof the star followed by a number. I don't know if we ever see a Roman numberal on screen, or if the characters merely give the planetary numbers in English..
When they discuss which planet to search in "Spock's Brain", the inhabited planets are described as the third, fourth, and sixth, planets, I think. And possibly they are numbered with Roman numerals or Arabic numerals on the screen.
Of course many
Star Trek writers were not well educated in astronomy nor science fiction fans, and so some of the star and planet names and designations used in various
Star Trek productions are rather strange.
The
Ex Astris Scientia site has a discussion of many problems with star names in
Star Trek.:
https://www.ex-astris-scientia.org/inconsistencies/bayer-names.htm
And remember that this has been a very, very short discussion of star and planet names in astronomy, science fiction, and
Star Trek.