I don't for a lot of the reasons already set out in this thread - "gamer" is mostly a synonym for folks who are antisocial at best but often much worse, racist/misogynist/etc - but also because there are so many different cultures in games now and I don't think most people are interested in all of them.
There's games that are so big as to have their own culture, usually major e-sports titles (DOTA, League, Overwatch, etc); there are folks who play shooters pretty exclusively; there are more mainstream indies; there are the indie circles that live primarily on itch; there is aaa console culture which is e3, game awards, etc; hardcore sports players live in their own world; old arcade stuff has its own scene and is even more so split between western and Japanese; retro fans & the preservation scene; and more and more, and not a lot of folks overlap all or even more than a few of these groups. It's like how a conversation may have started with "yeah I'm really into music" but you drill down into "well I listen to a lot of [genre]." It's okay to not see yourself in all of these groups and it's okay to not want to label yourself as such. I think it does a huge disservice to call it all "games/gamer culture" because it tries to dismiss the very real differences in activities and aims and how these people play, and I think it's a poor idea to identify everyone who is a part of one of these groups as a "gamer" for the same reason.
The folks who are both older and cling to that "gamer" identity tend to still carry a chip on their shoulder from an era when there wasn't the same kind of spread in cultures and the younger ones are trying to feel persecuted on some level by identifying as like 1/5 of the world's population. If it hadn't sprung from a perception of counterculture and edginess (and been loaded down with it more by marketers) I don't think it would be as negative-feeling now.
One other thing that follows from this before I go is that for all the crying and handwringing that happened when we saw the "gamers are dead" op-eds some years back - nobody in or around games looks at gamers as one big group anymore. Nobody. It's not how games are treated as a product or how they're marketed and it's been a long time since that's been the case. The identity doesn't even exist as a marketing segment for the industry because it's that split-up. It really is "dead" in terms of everything except shitty t-shirts from Kohl's.