I prefer to play vn's with gender choice, but i play bl's and gl's too. Very rare thing for me to play pure otome. I used to play everything years ago, but now there's just a lot to go through, my tastes changed, got pickier in general. Good thing there's a lot of novels where you can at least choose pronoun. As a not very gender conforming gay guy - that is as close to me as it can get. Do i ever feel self inserted? Nope. It's impossible and more like role-play for me.
When mc don't have preset looks or customization, i usually just rotate three names i always use. My own name for games set in present and close to reality, one name for medival/fantasy and one for anything else. Plus one name i use for sapphic/gl novels.
For mc's with set look's i go with canon names. Like, i can't imagine Palmier from Bewitching Sinners not being Palmier(god that dude's hot af, can't wait for Royal Blood).