This irritates me so much and is, in my opinion, an intentionally bad-faith take on the matter.
His being gay is not a factor in their relationship. It is a part of Dorian's story, and the Inquisitor standing at his side and supporting him through a tough time with his father is something any good friend—anyone who loves and cares about someone else and their wellbeing—would do.
Dorian's story is less about him being gay so much as it is a human story about acceptance, and the hurt that can come from a parent not accepting that child (to the point they try to force change). You could argue that's something most if not all people have felt or experienced in some capacity whether or not it has anything to do with who they're attracted to.
My character is a female mage. While our Inquisitors' backstories are somewhat up to us, mine was intimately familiar with the feeling of not being accepted by her family. In her case, it was magic. In Dorian's, it was who he was attracted to. They bonded not just over that, but in a matter of shared goals, a sense of integrity, and authenticity. They respect each other.
He is her best friend, but she would never view him as her "gay best friend." He's just Dorian. She values his opinion because she trusts his character, and vice versa. They aren't afraid to let each other in.
The only role his being gay plays in their relationship is in the fact it could never be romantic between them...and that's not a role at all. It's just a fact, like Dorian being a mage or from Tevinter is fact. It's one small part of who he is, but not the whole.
I don't understand how a healthy, believable friendship is considered "flattening."
Real friends open up to each other in tough times.
Real friends talk about life and relationships, with a few jests and friendly jabs in-between.
Sometimes, I think it's the fact that Dorian and the Inquisitor's friendship is so normal that throws them off and leads them to pursue a reading of it that doesn't exist.
Plus, I'll be blunt: Dorian's not exactly a stereotype in my opinion. He has the sass thing going and he's pretty (versus something like "rugged"), but that's about it.
He's one of the most beloved characters in the game for good reason.