Seasons are something of a culturally-ascribed overlay on the astronomical calendar, just as (e.g.) "evening" isn't a specific area on a clock..
I can't speak for all ancient calendar traditions, but one explanation is readily at hand from Gaelic tradition ...
The Beltane observation (early May) marked the beginning of "summer" in that tradition's calendar scheme.
This is something I have discussed with people before.
I dont know about other countries, but here in England the seasons starting at the cross-quarter festivals always made more sense that at the quarters.
At the beginning of Feb we get the first flowers, the snowdrops and stuff.
Around May everything seems to explode, the hedgerows go crazy. Everything is just growing huge.
At the beginning of Aug, thats it, everything seems more or less done. Everything is going to seed.
Around the begining of Nov its all over. Pretty much everything is finished. It all dies back, waiting for Feb when it all starts again.
I know that is very simplistic, but I can actually smell a change in the air around those days. An actual physical change, like one door has swung shut and another has opened.
I never understood solstice being the start of summer. To me it is the high point of the season. Summer at its fullest extent.
The same with winter solstice and the equinoxes.
People have argued with me, saying that jan feb and march are the coldest, so they must be winter, as though hot and cold are the definitions of the seasons. And I suppose that is fair enough, if that how they see it and how it makes sense to them. But for me the changing of the seasons is all about what the world is doing, and that smell in the air, that smells of spring, or summer.