Bear in mind, the terminal goal doesn't actually require unbiased numbers; the way most TTRPGs work is that you're trying to roll over or under a target number to get a weighted, unpredictable outcome. The idea is that while players (usually) want any given action to succeed, they some of their actions to fail in order to preserve narrative interest, while having their character be better at some things than others.

As such, while randomness is best, the given method is quite sufficient for having fun, and both players can agree that it's fair: they each have equal influence over the result.

I think it depends on the roll. For two players against each other it’s a bit more fun, but dm/player feels imbalanced, even when it’s character based. Th player winning might feel more fun beating another character with wits but that doesn’t work as well imo for luck, strength, or against something inanimate. It also moves success into a personal skill of the player vs the dm. I don’t agree that it’s necessarily fair.

Perhaps it’s just the feeling of “I took a risk and it didn’t work” vs “I chose badly and was outsmarted by the dm” seem different to me in an important way.

You could reframe this all easily as well for “I’m thinking of a number between 1 and 100, guess it within Y distance to succeed”. That’s mathematically equivalent I think, if you allow it rolling around.