You may be underestimating people's ability to bullshit their way through this sort of discussion.
It's harder to bullshit your way past a blank file in a code editor.
I'd wager that something like FizzBuzz will eliminate 90% of the chaff. Yes, it's laughably simple. No, it's not so simple that it won't stump a significant number of folks who've coasted for years.
I did a .. bit more complex than fizzbuzz but similar toy problem sort of thing recently, and it's nothing I can't do, it passed the visible test cases (HackerRank if you're familiar, I haven't used it before but I assume it's generally similar) etc. but it was clumsy and far from representative of what I'd actually write on the job I think. Even if I still only spent an hour on it, just not having that pressure. Plus I suppose I would likely have seen the problem before in planning, and realistically it would be something within the context I was familiar with from working on it every day, not the toy problem. (This wasn't it, but imagine something like calculating the possible states of a chess board after each player makes another move. Bit simpler though.)
Sounds like you'd more valuably/realistically get that from the discussion of a previous project though? How it works, or something interesting they had to figure out, etc.?
> don't be too hasty to think the people doing technical interviews are idiots
I don't think anyone has a problem with technical interviews? At least I agree that's not reasonable. That doesn't have to mean 'coding' though, you can ask about how they'd approach a particular problem, quiz on some fundamental knowledge in a fizzbuzz sort of way, etc.
You can more easily tune it to the kind of candidate you're looking for then too, for example I've asked how they'd tackle improving the performance of a particular SQL query that's been identified as too slow. There's a tonne of possible answers to that ranging from naïve/they don't really know, through pragmatic good fit responses, to way overkill I hope they understand we don't need them doing that here/not operating at that scale etc. - and it's fairly open-ended in what you can discuss driven by what they volunteer and know about. (Which is another good thing IMO, I don't like being on either side of interviewer quizzing and candidate umming and ahhing not really knowing! Both more comfortable and more beneficial to have a discussion about whatever is known IMO; to quickly move along the perimeter if you hit the edge of that.)