H7 actually surrounded himself with scholars that promoted the "New Learning" (Erasmus and More among them) and was a patron of the arts himself. While his mother was also an advocate for the arts and promoted education notably by founding colleges and patronizing devices like the English printer (Elizabeth of York too) as well as works of literature and translations. I don't think the Renaissance started outright after Bosworth, perhaps one could say Henry just laid the foundations for it.
You are right; there were humanists at Henry`s court. I overlooked them in the previous anon because many didn`t really like Henry. Thomas More downright hated him (which makes me rather like Henry) and Erasmus wasn`t exactly pleased what he found in England when he first came there in 1499 (or 1498) and was only induced to return after leaving in 1500, and joining court when Henry VIIII ascended the throne. So Henry was not exactly a favourite of them, and I kind of ignored them because of that.
Two scholars Henry VII did keep around him were Polydore Vergil, whom he paid to write a History of England favouring him (which it very much did, but which nonetheless in places written after Henry`s death sharply criticised him and his greed) and Bernard André, who arguably adored Henry and, as @lizzie0278 points out, basically wrote fanfiction about Henry. Henry reportedly loved his poems, which tended to favour himself. That`s not really so much an indication of an interest in scholarship as it is an interest in self-portrayal. Which there is nothing wrong with in itself; it was an important skill to have for a king at the time. It is just not all that relevant for this argument about the Renaissance.
As for Margaret Beaufort patronising education by founding colleges, this is true but not overly notable. Usually queen consorts did that, and we have evidence Margaret of Anjour, Elizabeth Woodville and even Anne Neville, about whose tenure as queen we otherwise know very little, did it. The notable thing about Margaret Beaufort doing it is that during Henry VII`s reign, she did it and his wife Elizabeth of York did not. This, of course, squares what has been said several times about Margaret taking over many of Elizabeth`s traditional consort duties.
As for the English printer, that may be correct too - I have neevr read anything about it, but maybe I have simply forgotten - but it was, at that point, hardly notable anymore. In Edward IV`s reign, William Caxton and printing in general were favoured a lot. Especially the Woodville fraction did a lot to see to this being done; Elizabeth Woodville`s brother Anthony wrote one of the first books, if not the first book, printed in England.
Richard, too, favoured books and printing in his reign a lot, and Caxton decidacted a book to him for that - which was a gesture, but hardly a gesture he would have done had he not known it would please the king. Richard`s legislation favoured the import of books, and Richard was also the first king who made the law accessible to a broader audience of people by having them written in English and thereby enabling everyone who was literate to read and understand them.
It is not that Henry did do nothing that was expected of him, but he also did nothing special at all, or nothing that had not in most cases been done before and more thoroughly in regards to education. So to say he laid the groundwork for the Renaissance is either ignoring all that went before 1485 to make Bosworth the cut off point for which actions regarding the Renaissance suddenly counted and all that went before, even if Henry only continued it, does not matter, or else the idea that Henry brought the Renaissance to England or even enabled it through his actions does not have much to stand on.