Okay, so the first thought I had, having sung a few choral works from the Renaissance is that we may want to back up the truck and ask not why a major sixth was considered a dissonant interval for "harmony" and ask why it may have been viewed as a dodgy interval in melodic writing. It's been, I admit, decades since I have sung Renaissance vocal music and just as long since I fielded theoretical reading on that topic but ... I remember noticing that major sixths were not common melodic intervals.
Now imagine, if you will, that you are in a vocal ensemble performing music and you get your starting pitches but you have no instrumental back up or if you have instrumental back up you are singing with the notes (scholars have debated whether this is legit but I don't want to rabbit hole into historically informed practice and how Taruskin and others expressed doubts about how HIP the HIP could really be). Let's bear in mind that many of the singers may not be professionals and some of the singers may be professionals but there may be kids in there (boys singing parts that would eventually be sung by women or girls, not counting castrati). Imagine that, with no back up for your pitches, you all have to sing a descending major sixth. How many of you have done this? In my experience large descending intervals like sixths are perilously easy to screw up.
The warnings against major sixths in Renaissance treatment may, if memory serves, reflect some practical realizations that there are intervals that may be so badly bungled so often by less-than-ideal-singers it was better to warn composers to just not use those intervals. Vile an loathsome discords can have practice-based reasons that are not "just" related to tuning systems, although surely tuning systems have played a part. Even with college choirs if you have enough melodic sixths that get bungled by music majors a piece that was cued up in C major by the choir director at the start can end in A flat major even though there are no modulations specified in the work and no scored changes of mode. I am not speaking hypothetically. I was in a choir that screwed up in exactly this sort of way and I think it was some kind of Romantic era choral work. Maybe.
Renaissance rules about vocal writing put a lot of restraints on melodic writing practices that can seem opaque and absurd to us because we're used to choral music where everyone has accompaniment and back up notes or can listen to practice tracks. Imagine you have none of that, that you have to be able to sing what's on the page under not-so-ideal conditions, and that you got one shot at this. Do you want to have some major sixth in your melodic line then? Eh ... I hope you're getting an idea of why certain intervals were carefully handled in Renaissance music. Bear in mind I'm not a professional musician and I'm not claiming more than rusty first-hand experience singing Renaissance era music, but I am thinking back about how, yeah, major sixths were handled with care. It wasn't that you could never use them, it's that they had contextual preparations.
If I wanted to end this with some pseudo-Marxist flourish I'd say the material means of production were hugely important and this would mean not just the tuning systems but also the sheer limits of the not-always professional human singing voices that had to perform this stuff.
