It's slow to run, and needs a long configuration then calibration time until you don't find false positives anymore.
Also, many bugs found by fuzzing are in the "it's ok to have them" range.
10Mo URL crashes my Python interpreter? I'm ok with that.
Not to say fuzzing is not useful, but it's not cheap, and the benefits really shines for mature software.
Either you're wrong about the "false" part or you're testing the wrong entry point.
Fuzzing has a long warmup time but probably the lowest false positive rate of any testing technique - if it crashes you're wrong about some invariant, even if that's only "I need to fuzz parseX, not decodeX".
A thing is only a bug if it's a problem for use case your users will have an issue with.
The first couple of runs everything breaks, then code is added to NOT catastrophically fail, then you are pretty good.
(or move into lots of false failures)
Sort of like -Werror or linting/static analysis type tools, or code coverage, or similar...