"to weed out the vast majority of applicants that can't code at all."
I think one ought to consider the possibility that if the vast majority of interviewees can't code at all, there is something wrong with the company/process. I mean, I'm not saying it necessarily follows, but the question is worth considering.
I have never been asked to implement FizzBuzz, so I can't buy the claim that "everybody does it".
Rightly or wrongly, if I was asked tomorrow, I wouldn't have the preconception that it was appropriate or necessary, and thus wouldn't necessarily choose to continue if I thought it was indicative of culture. If other people react similarly, it doesn't necessarily matter if it is a reasonable standard of competence; you're still skewing the pool in other ways than you intend.
I think FizzBuzz is an example, when I was interviewing I mostly got questions in this vein during first or second telephone screen (some companies use an online multiplayer notepad or some other method).
When I was doing interviews I was always mostly discussing motivation. Most of the time the hires turned out very good, but I did happen to have one bad one and it cost us quite some time.
I have had the experience of doing well on the programming test at the beginning of the process and being rejected for other reasons. Maybe counter-intuitively, that made me less interested in doing such tests, since in that case it wasn't determinative. It's possible that many or most places feel that to ensure someone doesn't cheat on a take-home, they must demonstrate their ability in the in-person interview. Which is logical, but it also makes me think that going through a process starting with a take-home is a waste of my time, as long as it's not a universal requirement.
What you describe is normal. Assignments usually go from easy to hard because usually it is faster to solve the easy problems and usually these questions can be also asked by less technical people. Several companies I have interviewed with had the first programming questions asked by recruiters (who clearly had CS background, but most probably just shifted careers)
What I'm talking about had a much more involved test than Fizzbuzz at the beginning, IIRC something like 5 small projects, but with your choice of language, and a generous amount of time (days). During the interview, it was simpler, but in an environment I'd never used before and a language which I had disclosed up front I didn't have recent experience in. It seemed like they were only interested in hiring new college grads at the end, but perversely the process started by senior people referring me and screening me. While I admit it sounds like sour grapes, I thought later that the organization probably had communication/coordination problems that would've made it unpleasant to work for.
In some organizations, they start by recruiting people they think have talent in general, and then the interview is about finding where you fit in.
In others, they have a very specific need and they focus on winnowing down the pool to get the (hopefully) best person for it.
The place I'm talking about seemed to have a split personality.
Also, I've had enough interviews, both successful and unsuccessful, to be clear that there is no "normal". Every place is unique, as was this.
I think one ought to consider the possibility that if the vast majority of interviewees can't code at all, there is something wrong with the company/process. I mean, I'm not saying it necessarily follows, but the question is worth considering.
I have never been asked to implement FizzBuzz, so I can't buy the claim that "everybody does it".
Rightly or wrongly, if I was asked tomorrow, I wouldn't have the preconception that it was appropriate or necessary, and thus wouldn't necessarily choose to continue if I thought it was indicative of culture. If other people react similarly, it doesn't necessarily matter if it is a reasonable standard of competence; you're still skewing the pool in other ways than you intend.