You've convinced me that the dynamic selection of backstory-related tags, as the game progresses and the player chooses dialogue options, wouldn't be such a good idea. Indeed, PCs that are lying or are amnesiac would pose difficulties. To which there are two main solutions. One is to just let the player know that secretive or amnesiac PCs simply aren't supported. Another one is to not use this method. Which means : mini-questionnaire at character creation is the way to go ... Possibly. I want to emphasise 2 things though.
Firstly, this questionnaire should really be minimalistic. The first 3 examples I gave are probably sufficient.
- Place of origin. The list should include one entry for each place for which there exists content (i.e. at least one person, somewhere in the game, who has previously met your PC). If the player writes any "NameOfPlace" of their choice, then it means they won't meet anyone known to the Pc, they'll just be able to have this place written on the character sheet and have the PC say "I'm from NameOfPlace".
- Previous experience with adventure or danger. This could perhaps allow for slightly different party dynamics. Maybe your companions would be more impressed if you prove surprisingly adequate in adventuring when your PC never did this before. And maybe they would be more ready to defer to you if you've let them know you have some experience (well, Astarion and perhaps Shadowheart).
- Age. This could give access to various exclusive dialogue options. Or ... just not exists at all. Well, it would be nice on the character sheet, even if it has no impact.
It is completely clear that Larian cannot (and should not) acknowledge in some form, and provide reactivity for every backstory that players could possibly come up with.
Secondly, all this is really
if some backstory elements have to be used at all. And I don't think that, for a Fully-Custom Character, the backstory should be a main point of focus. What should count about the PC is who they are and the choices they make. What they are made of, what they really value, and who they become.
Hmm, limiting character roleplaying possibilities by telling PC that they cannot lie or be amnesiac would be faulty, in my opinion. I'll agree.
I also agree that for the sake of avoiding potential clashes and player limitations that the questionnaire should be kept to a minimum, but the answers should still grant some flavor content, similar to that of the Origin characters (like discusses earlier). And I do like the 3 qualities that you brought up. :] Quick, clean and efficient. Does not require too much studying on the player part either.
And I agree (... yet again) about your second point. Trying to involve the past of a FCC too much is simply gonna put restrictions on the FCC in the end - and I am the kind of person that is willing to trade small flavor interactions for my blank sheet. I do so very much love to make my own story, as detailed as possible, even if I cannot "use" it ingame in any way.

Can you explain why you think FCC is not already in the game?
Pretty sure Darth Malorn pointed on on multiple occasions that current Tav is pretty much a FCC. *BUUUT*, one could argue that this is false since Larian presumes that the character we play as is from Baldur's Gate (and, to my greatest sorrow, it seems like this is intentional and indeed made to stay :[ ).
Like Malorn said, the entire point of a FCC is that NOTHING is presumed about them. Everything is in the hands of the player, but at the cost of some special interactions and that NPCs might recognize us. In BG3, Larian DOES assume some things about our character - like Tav being from Baldur's Gate, and that makes them a SCC.