The mission of the NX-01 Enterprise did not lead to the birth of Starfleet.
Starfleet already existed for 20 years when the NX-01 Enterprise launched.
The NX-01 Enterprise was a Starfleet ship.
The mission of the NX-01 Enterprise lead to the birth of the Federation.
The Star Trek Picard season 3 writers continue the proud tradition of NuTrek writers not knowing the difference between "Starfleet" and the "Federation".
"Federation Headquarters" LOL.
This is so bad nitpicking I thought it had tobe a joke for a moment. Okay. Sit down... time for a lesson.
Sooo, what year was the US Military, which Starfleet is most directly based on, founded? Well turns out the US Army says they were founded June 14th 1775. It's even on their flag. They proudly say they were founded before the US But is that ACTUALLY true? No it's not.
The Continental Army was founded June 14th 1775. The Continental Army was disbanded (after some debate about having a standing, regular national army) in 1783 after the US won the Revolution. What followed was a series of standing armies that came and went. The first was the Legion of the United States in 1792-1796, which was renamed the United States Army. This was followed by the Provisional Army of the United States from 1798-1800 that operated along side the US Army until it was disbanded.The US Army started to take a more instiutiionalized shape after the end of the War of 1812, which saw the US unable to defend itself within its own territory from the British.
Organization of this institutionalized US Army was an uneven thing for much of the 19th century. It leaned very heavily on State forces before and after the Civil War, consistent with the very federalized nature of the country at the time. The standing Federal force was uniformly very very small and only had combat potency because of those State forces.
This started to change in the 20th century. Technological and political changes in the 1900s and 1910s in advance of World War I lead to a reorganization of the US military that was happening simultaneously to that process occuring in European countries. The Federal United States Army really grew as a consequence of conscription in World War I in 1917 (the National Army) and after the war was glided into a smaller, but larger-than-ever federal standing force, that became the framework around which the enormous "Army of the United States" formed for World War II. It is that Army, when demobilized, that formed the basis of the post-World War II "United States Army" that has a direct lineage to the present.
So when was the US Army founded?
->1775? (Continental Army)?
-> 1796? (Legion of the United States, aka United States Army)
-> 1814? (Reorganization into a truly national force)?
-> 1917? (Birth of the National Army, that created an organized national army as we think of it in modern terms)
-> 1947? (Reorganization of the US Defense establishment into a sustainable post-War national standing model)
The answer is all the above, because the Army is an idea and an instrument of the state, and not a person. It could never have one birthday, because the current idea of what an Army is, is barely 100 years old. It certainly isn't nearly 250 years old.
Now let's consider Federation Starfleet and what we know of the creative intent behind Star Trek: Enterprise. We know that Earth Starfleet is not Federation Starfleet. But we do know that the creative intent behind Earth Starfleet was that it would become the backbone around which Federation Starfleet would form as the Tellarites, Andorian Imperial Guard and Vulcan fleet were integrated into it. Earth, as Enterprise made clear, was the only one trusted by all three. So in a sense, this is even a simpler path than Continental Army-> US Army. And that is what you are seeing in this episde. Just as the US Army claims its creation on the date of its factual direct prdecessor that it has no direct linkage to, so to now, does Federation Starfleet. It is a distinct entity from Earth Starfleet, but Earth Starfleet is the heritage it claims. So what you see, what Shelby did, wasn't a mistake or "NuTrek" forsaking established precedent. It was ENTIRELY in line with creative intent of Enterprise in 2005.
And let's just recall, to further the point, that the Coalition of Planets, created in Season 4 of Enterprise and as the Government that, in the Enterprise retcon, partially (but not entirely) fought the Earth-Romulan War, was designed by Enterprise producers to be analogous to the Articles of Confederation-era US, or League of Nations era international community before the United Nations formed. Or perhaps more relevantly, the United Colonies that predated the United States in 1775. It was designed to be a looser, prototype political relationship, that was a stepstone on the road of to the United Federation of Planets. In a very smart and accurate real world sense, the species of local space would have to discovery how inadequate the Coalition of Planets and relying on 6 or so different fleets was, via the Earth-Romulan War, before in the wake of their victory, consolidating politically and militarily into a stronger, far tighter alliance. Does that sound familiar? Because that is what the first 25 years of America's founding was, more or less. Enterprise Producers were telling that story, but for the Federation.
So really the mistake is yours here, in thinking that you are the authority on canonicity on what Federation Starfleet is and is not from a historical perspective in terms of heritage claimed (not in terms of literal specific institutional founding). That was never established firmly in canon, because Enterprise was the last series that clearly dealt with the issues of Federation founding and political relationships in that period. But now it has been. Star Trek Picard, via Shelby, just created new canon for you: Federation Starfleet claims the heritage of Earth Starfleet as its direct predecessor, and like the United States Army, claims its founding predated the founding of its government.
We should be happy this is the case, because this is an interesting worldbuilding detail.