By the end of endgame? No they control less than maybe 15% of the galaxy. By the 29th century and given the rate of fed expansion maybe.
IMHO you are praising the Federation with faint damns, so to speak, by saying they control less than maybe 15 % of the galaxy. I would think that they rule a much smaller part of the galaxy.
Do the math. E.E. Smith did in the
Lensman series where two rival realms each rule most of a large spiral galaxy. Thus each realm rules millions of planets and has space fleets containing millions of space battleships.
But in
Star Trek the United Federation of Planets has 150 full members. In 'Metamorphosis":
COCHRANE: Believe me, Captain, immortality consists largely of boredom. What's it like out there in the galaxy?
KIRK: We're on a thousand planets and spreading out. We cross fantastic distances and everything's alive, Cochrane. Life everywhere. We estimate there are millions of planets with intelligent life. We haven't begun to map them. Interesting?
If Earth Humans have 1,000 colony planets and each of 150 Federation Members has as many, that makes 150,000 planets in 150,000 star systems ruled by the Federation. And maybe habitable or otherwise useful planets are very thinly scattered and there are only 1,000,000 habitable or otherwise useful planets in the galaxy, making the 150,000 in the federation 15 percent of all the ones in the entire galaxy.
But consider how many star systems are within a specific radius of Earth, and how many of them have Earth colonies or Federation member planets. Thus you can calculate the approximate percentage of stars within a volume of space that would have Federation member planets or Earth colony planets. And thus you can calculate how large a volume of space the Federation would have to rule to have 150,000 star systems. And it would pretty much appear like a dimensionless dot on a map of the galaxy.
In "Balance of Terror" McCoy said:
MCCOY: But I've got one. Something I seldom say to a customer, Jim. In this galaxy, there's a mathematical probability of three million Earth-type planets. And in all of the universe, three million million galaxies like this. And in all of that, and perhaps more, only one of each of us. Don't destroy the one named Kirk.
So if there are three million Earth-type planets in the galaxy, and all 150,000 federation planets are Earth-type planets, the Federation should rule about 0.05, or five percent, of the volume of the galaxy.
Cochrane asked what it is like in the galaxy and Kirk said they estimate there are millions of planets with intelligent life. If there are between 1,000,000 and 10,000,000 planets with intelligent life in the galaxy and there are 150 separate intelligent species in the Federation, the Federation rules about 0.000015 to 0.00015 of the intelligent species in the galaxy, and thus about 0.000015 to 0.00015 of the volume of the galaxy.
And what does science say about the number of habitable planets or intelligent species in our galaxy? Back in 1964, Stephen Dole of the RAND Corporation tried to calculate the number of planets in our galaxy habitable for Humans in
Habitable Planets for Man. He estimated the number of habitable planets in our galaxy to be be much higher than McCoy's three million Earth-type planets - about a hundred million if I remember correctly.
Here is a 2013 claim that there may be 2,000,000,000 planets in our galaxy suitable for life.
https://www.theguardian.com/science/2013/nov/04/planets-galaxy-life-kepler
Here is a 2014 claim that our galaxy could have 100,000,000,000 planets that could sustain complex lifeforms.
http://news.cornell.edu/stories/2014/06/milky-way-may-bear-100-million-life-giving-planets
If one out of every thousand to one out of every million planets suitable for life has intelligent life, then for each billion stars with suitable planets in our Galaxy there should be between 1,000 to 1,000,000 planets with intelligent life. Thus if there might be 2,000,000,000 to 100,000,000,000 suitable planets in our galaxy, and if the estimate that between one a million and one in thousand of them currently had intelligent life is correct, there are presently between 2,000 and 100,000,000 planets with intelligent life in our galaxy.
So with 150 intelligent species, the Federation would rule between 0.075 and 0.0000015 of the volume of the Galaxy.
Of course that is just a wild guess based on rough scientific estimates, but it shows that 150 intelligent species might possibly inhabit only a very, very, very tiny part of our galaxy.