Picking the bit you like makes it hard to reason about how secure that is. It could be your preferences are much narrower than you realise. If you're going for something you find aesthetically pleasing or memorable then that's probably going to seriously limit password entropy. On the other hand if you're trying to pick something that "looks random" then you should know humans are terrible at that.
20 perfectly random characters from a base64 stream have 20*6 = 120 bits of entropy. If you pick 20 characters from out of 1000 choices you can see on the screen, that would seem to remove at most 10 bits of entropy, no? (2^10 ~~ 1000)
"picked a bit I liked" means selecting a length suitable for the site without any weird characters that the site will reject, not trying to find my place of birth in the output!
I dunno, I wouldn't discount this entirely. An advanced targeted attack could likely profile a user for their "likes" in a random string (e.g., favorite letters appearing often), but in the case of anonymous, brute-force-style attacks, "I just click things and scroll around until I see something I like" is probably really useful. It could, for example, protect against a somewhat predictable generator, since you won't know "the user always uses the first password, and the first password is always seeded incorrectly", or some other specific implementation flaw.
