I was thinking about this the other day and found this forum so I signed up to answer. Hopefully the OP is still around. I skimmed through the answers and have formed my own opinion/theory which may not be exactly correct, but I believe it is a little closer.
The short answer is that the letter names were assigned to the notes first, based on the lowest note that they used, and eventually the scale which is now "natural" to us, C major, actually just became the most popular or pleasing to us.
This article (
https://www.ars-nova.com/Theory%20Q&A/Q65.html) is a little bit of a different question but I think it provides insight into this answer also. Basically what the article states is that Boethius, as other posters have mentioned, started the convention of naming the notes of the scale with Latin letters, starting with the lowest note that they thought of back then being called A.
I don't think we know exactly what kind of scale Boethius was thinking of when he started on A, but if you think that this is in the context of sacred music you can sort of see that he may have used a more minor like scale, maybe Dorian or something similar. When I think of sacred chant music a lot of it uses those natural minor type scales. Cudo's explanation above, of the Hypo-Dorian scale, would seem to make sense, although Guido of Arezzo did not introduce the letter names. What Guido did was to develop the solfege system of learning to sing and read and be musical. What the article above states is that he developed the solfege letter names (Ut (later changed to Do), Re, Mi, etc.) from a well known Latin Hymn called Ut Queant Laxis. Each new line of the Hymn starts on the next note of the scale and the Hymn starts on note C so C became Ut (Do). This is much like the Sound of Music song "Do a deer, a female deer, Re a drop of golden sun..." etc. (I wonder if Sondheim knew all this history? Probably)
Anyway, what I believe is that church music used many minor type scales in their hymns and that minor type scales were probably thought to be lower or first and so when they named the first note of their scale with a letter they started with A. It is all a little arbitrary, really, but it does make sense to start with A, rather than start A on the third note and have the two lower notes below that be F and G. Which wouldn't make sense for Boethius because he didn't actually repeat A to G but kept just going on up the alphabet from the lowest note A up to the highest note which was O or H.
Eventually, the scale which started on C became more popular and memorable for people, to the point where Guido used it for his system which probably solidified the idea that in a key with no sharps or flats the scale starting on the third note is C and it also probably solidified the C scale as the more popular scale.
This was all way before the modern keyboard configuration, I believe, and so it was already established that C major had no sharps or flats, so it wouldn't make sense to have all the white keys represent A major, since, for example, C natural and F natural would then be black keys!
Hopefully this helps!