TStael: Gaming immersion matters - be this GTA5 or DA. I will never go for the former, but I expect to have the latter.
Do you really think there is a "sole = proper" way to game? And that "best" way excludes anyone?
Immersion first of all requires a certain measure of openness on the recipient's side. And IMO gender is only a minor "obstacle". I have no problem playing either male or female characters (in fact I will often choose the latter if possible), good or evil, loyal or backstabbing, lawful or "chaotic" (again, preferring the latter)...
There are a few immersion breakers for me, but they are entirely my concern and I would never demand developers change them to suit my needs. For instance I can't stand wanton cruelty against animals - so I'll never use the "silencer" in Postal 2. And I feel awkward romancing a male NPC, no matter if I play a male or female character. But I see those as "blockades" in my own mind, not as "exclusion".
If you have such a hard time getting immersed in games that's characters "aren't you", IMO the problem is with yourself. Do you have the same problem with other media too? Can't empathise with a character of a movie or book because they're very different from you? If yes, you should really work on yourself, because you are missing
a lot. If no you should maybe try to find out why it works in some cases, but not in others.