I think that's more a writing problem in general.
I mean, the thesis is problematic here as it assumes playing as someone else is limited only to skin color or orientation, when the reality is, with the exception of actual soldiers playing call of duty, there probably isn't an example of any game character that resembles the player to any great detail. I mean, yes, visually I'm a straight white male, but how many games star 30 something office clerks with minor weight issues? We play precisely for the different experience: to be the soldier, the warrior, the adventurer, the survivor, the hacker, the alien, anything we aren't.
The experiences you talk about have to be part of the narrative, not the external features. I can play an Asian character easily enough by slipping in a dynasty warriors game, but there's no part of the game where that impacts the experience. People seem to clamor for a female link, but to me it'll just be the same poorly written justification for a fetch quest with a different batch of polygons. Hell, what was the Assassin's Creed thing about: fucking multiplayer where there is no story and appearance literately changes nothing. Meanwhile the opposite can occur even with white characters: wondering if the gang lords want to blast your lily white ass in Far Cry 2, or being an elf or mage of any color in Dragon Age, or being the upstart human (siding with the dangerously pro human movement) in Mass Effect.
Since most games' stories and characterizations exist purely to move the plot along, nuance like that even getting attempted is rare, ultimately reducing diversity to the pallet swap that separated Mario and Luigi at first. What you want requires two tings: the first it fixed characters. We can't have character customization and have sweeping differences to how the story plays out based on appearance and orientation without a metric fuckton of extra writing, voice recording and programming. The other is talented writers that can portray such nuances without coing off like a captain planet script, or "pity me" self insert fanfiction. Cultural politics stops informing and starts being humorous if you deal with one dimensional overt prejudice. Part of what mae the Lee / Larry dimanic work that way way is could be racial, but is also might not have been and as the player, do you take it personally, or you you realize you're a recently convicted murderer and someone that knows that has every right to pre-judge you.
Just to end on an example: in comics I liked the character Steel one of the replacement Supermen after his death. He was black, but he grew up in the projects, lost his parents during a demonstration and brother to gang violence, and had a unique view of the world and life that impacted the book and how he handled things. You don't get that by just making Clark Kent black and telling the same story trying to slip in a scene of racially based bullying.