When I've seen people advocate Fizzbuzz, it's always in a context of how 90% of their candidates can't even do it. It's intended as a quick screener that immediately eliminates a ton of obviously bad candidates.

You're clearly getting a better caliber of candidate than those people are. That's great! And clearly, if that's the case, you should adjust accordingly. I think the lesson there is to tailor your process to the people you get.