As much as I hate to admit it, I do think that if you were to of executed these characters as "Heroes" or "Legends" that are named as if they were closer to something like borderlands, Instead of making them "Player Classes" and such, it would actually make it more digestible. At least in my experience with gaming. I can't explain exactly why my brain decides to work like that, but for sure it at least helps me dissociate myself from the character I'm playing. I don't think this would be preferable, but if you were unable for whatever reason to unlock gender, it wouldn't be the worst path to take either.
I think the best example for me personally is Overwatch OR fighting games. I think there's something about making them proper named characters that would help re-contextualize it for my brain. It feels more like i'm working with this character rather than becoming this character, which it might just be a me thing but I often naturally fall into trying to become my character, even when it's not an RPG.
The voice thing wouldn't help. It might help a little bit, but not by much, bc I know it's still there. I'll try it though sometime lol, maybe i'm wrong. I just know when I try to pretend something isn't or is there, it doesn't help a lot.
One other thing I want to mention, is that there was a recent game called "Lunacid" a kings field inspired dungeon crawler. It has a wide range of selectable pronouns that are never actually mentioned in game or referred to the player as those pronouns. It's just another section on your character sheet, it's just fluff. I wouldn't mind this being in more games, because it really does mean a lot to me to have that control over my character.
Maybe it IS just placebo, but just having some tangible proof, such as the pronoun example, goes a very long way. I know you'd rather fully implement a character customization system before just adding pronouns and calling it a day, but it would help me a lot personally. it really does feel like a nice minute detail/control/fluff over your character, and it wouldn't hurt to add since you're already avoiding gendered terms in your script anyways for the most part. Just a suggestion ofc for consideration after you feel happy with character customization down the road.
Totally understand that it's just not in the cards for the near future due to budget restraints, I appreciate it even being considered.