My #1 complaint about ASIs is that it's either them or Feats. I want to take the Feats, but my innner optimizer makes me take the ASIs. Boring.
I think it's funny that the very first response to the question is the 180 opposite of my position. They only get to stay if they share space with something else, and I happen to think feats is the perfect partner.
My bias, like lowkey13's, comes from "growing up" with AD&D, et al. Having any ability score, ever, improve was a really big deal. Sure, there were age-related trade-offs, but most PCs either died or retired before they became a factor unless an elder character came out of retirement to bail out a young-un. Though I was generally receptive to the 3E changes, this was something I did find objectionable. I've gained a certain amount of tolerance for it, over the years, and can see where it makes sense for characters to actually improve in that way. It's still not something I want to see a lot of.
Even though I initially embraced feats in 3E with enthusiasm, I found the reality to be somewhat less awesome than I'd hoped. Feats were one of the main "game mastery" levers in 3.5 and the abundance of them (both in terms of slots and the literal thousands of options) drove the game to the point where Fantasy Hero started to look less more accessible for casual play. Don't misunderstand: I
love feats from a customization perspective, but really loathe them as something for players to look forward to for "power-ups".
So, the reduced rate of ASIs in 5E, along with having to share the slots with feats does a pretty good job of putting both where they really should be, IMO. I get a bit nervous when I see feats in a book or UA column, but things are OK, for now. Let's see how Xanthumgum's Guide to Everything goes and then ask me again.
If I were going to make any changes for a hypothetical 6E, it would probably be to change ASIs to "+1 to a single ability" and then balance feats against that fairly tightly. Which means lightweight statistical modifiers packaged with thematically strong ribbon abilities. Actually, I'd be really interested in seeing ASIs
as feats such that you only get that +1 Strength if there's a thematic ribbon that goes with it. Because feats can't generally be taken twice, I also like how that would keep the overall stat scores in check. The expectation of having a 20 in your primary stat is just silly.