Do y'all (on both sides) view choice of character class the same as race? Sub-classes? Alignment? Deities?
I know I do.
I've overtly banned evil PCs from my games due to past inter-party issues. At the same time, took one player aside and let them play a Lawful Evil character with the understanding that the character would only be allowed as long as they worked co-operately with the group. The character lasted to 6th level, before dying in battle saving the party. The group only found out he was evil as he uttered his dying breath, "You idiots, we could have made a fortune..."
I've also, as DM, restricted classes and subclasses - but usually for mechanical reasons. Back in 3/3.5E, there was a dragon mage class in one of the official books I made off-limits, because in my homebrew the background for the class didn't fit (it got its power from a sort of dragon worship/bond, which was something that Amberos dragons wouldn't stoop to allow); I relented when a player came up with a really interesting story/background, and that character ended up being a really interesting addition to the game, and well-liked by the other players to boot.
When it comes to deities, I like to stick to what's available for the campaign world - no worshipper of Pelor in FR, for example. However, my homebrew has not only particular gods of its own (Dhorian, Ziga, etc.), but even older religions such as Egyptian, Norse, etc. for the gods that "came before". Because of plane-hopping, it's had one or two priests from non-native deities (such as a Pelor priest), who could still get access to their spells, and even tried to set up a church in the realm (which the New Gods took a dim view of).
So, I've been on both sides of that fence in those cases, and I
try to lean towards being open to at least give it a chance and see if it will fly in the game. Sometimes, it works out - other times it doesn't. Them's the breaks.