Then I think it's wrong for you to say it was "not that either" as if you had canon facts to the contrary. After all, I only said I had heard that, and I have, and you don't have canon facts to disprove it. Though you could certainly say the assertion is not canon fact.
However, I would surmise Andorians, Tellarites, Vulcans and most other Federation citizens wishing to serve in Starfleet do not go the Earth's space academy, or Starfleet academy on Earth, but have their own schools - at least at first, possibly like undergraduate work, or a year or two of general education. I'm fairly sure all Federation academies would adhere to certain Federation principles and agreements and require minimum standards, but I think it somewhat arrogant to assume everyone wishing to serve in starfleet has to come to and go through Earth's Starfleet academy the whole time and nothing they have learned at home counts. Though I do also suppose they could take entrance exams and be placed in more advanced classes and skip the stuff they demonstrate a working knowledge of already, too, but if so, then they didn't really go all the way through Starfleet Academy, like Spock did.
We've never really seen these schools on other homewards, but I'd wager they'd be far more home-race-centric, just as Earth's doubtlessly teach much of Earth's history and human ways of thinking. What we share in common, Federation ideals, of course, would be required course work in all Federation schools.
Now maybe other races have to come to Earth, and maybe every Federation member is fine with that, but it seems wrong and unlikely, and ultimately, impractical.
We do know non-Federation members have to be sponsored, and I suspect they'd start going to the home world schools of the sponsor before moving on to more advanced training on Earth, assuming they don't wash out in the first two years of general education.
But the thing about Spock is he left Vulcan early on, mostly due to his half human heritage and resistance to Vulcan snobbery (if some are to be believed), but it was against his father's wishes when he chose to go to Earth and Starfleet academy there, instead of the vulcan Science Academy, so he could well have been the first to do that.
As for the crew of the Intrepid, I'd expect most of them graduated from The Vulcan Science Academy (VSA). After all, all Vulcans who had ever qualified to go there did go there, until Spock. Are you saying no one aboard the Intrepid qualified to go to the Vulcan Science Academy? Or they did, but just went to Starfleet when much older, having already graduated from the VSA and learned most everything Starfleet Academy could teach them anyway, at least for the beginning 2 or 3 years of a 4-year stint?
Well, it's not like I have canon fact to prove any of that, either, but I don't find it odd or unlikely Spock was the first vulcan to go to Starfleet academy on earth, go through the entire 4-year course, and graduate. And then he was under the command of Captain Pike for something like 11 or 13 years? Anyway, there would be plenty of time after that for hundreds of vulcans to go through the academy on earth for the full 4 years, too, if that's an absolute requirement, and still fully man a Constitution Class Starship permanently assigned to the VSA and the planet Vulcan. I'd just prefer the assumption most of those simply went through and graduated from the VSA to qualify for a berth on that ship. If they had to finish up at Starfleet Academy on Earth, then so be it, but to demand they do the full 4-years regardless of what they already learned elsewhere is just so much harder to believe. And for officers on the Intrepid, I bet there would be no problem using experienced members of the Vulcan space fleet, whatever it was called, and many of them may have never stepped foot inside the Starfleet Academy schools on Earth.