This is strictly dependent on the GM/Player pair
It's a bad idea if either the player or the GM are not interested. It can be a great idea if both are interested.
I have done this, but as a GM rather than a player. I had designed a setting where it would be highly beneficial to have at least one character with directly-plot-related amnesia. I could, in theory, have made it work with all the players having amnesia-- in some respects it would have worked better; in other respects, not as much.
I pitched the idea to the players: I got one (maybe two?) vehement refusals to consider the idea, at least for their characters. The one I remember clearly was not so much a trust issue (I hope) so much as that player loved creating their own backgrounds and felt like they would be getting robbed of that experience.
I got one enthusiastic expression of interest. This player told me at the time that this was great because they were just too busy to put the amount of effort in on a character... but, they had some conditions. (I would not make them the secret bad guy, I would not put them in a situation where they were doing something gross like accidentally romancing a family members, etc, etc. I of course agreed to this conditions.)
The remainder were sort of, "Meh, I'll take one for the team if you really need this for your game, but I'm not excited about it."
If someone approached me in a different game and said they wanted to play an amnesiac character, then unless there was something about the game or setting that made this hard to handle, I would certainly allow it, but only after talking to the player to make it clear that they really are giving me permission to create their own hidden backstory, and also to find out if they have any lines they don't want crossed or buttons they don't want pushed.
And if I had the chance to play under a GM I knew and trusted, and they asked me to play an amnesiac, I'd probably do it. (I probably wouldn't suggest it, because I also enjoy coming up with backgrounds for my characters.)
Who does this?
Players who don't like coming up with backgrounds or are just too busy in their day to day lives to come up with backgrounds.
Players that trust their GM. Like, really really really trust their GM.
I've never seen a GM react negatively to this, but one possible reason why is that they took it as a slight to the background of their world. ("What, my world is boring and you don't care enough to make a character background?") That seems excessively touchy to me, but it's possible. They may also have taken it as a sign of disinterest in the game as a whole.
If you put this to a GM as an expression of trust, and also point out that this gives them a way to hook your character into any background plot that he or she thinks is not getting enough emphasis and that you promise to hold up your end by caring about this and investigating this... that may change their considerations.