I agree, if you know your requirements are such that you don't care about the correct answer, but any good enough consistent answer will do (e.g. the fast inverse square root), then doing the full numerical analysis is not worth your while (though I'd still do some back of the envelope estimates just to make sure the answer is in the "good enough" category and not in the "wtf" category). The issue is when there's libraries involved (especially more generic libraries are involved), typically they don't document what assumptions they are making, so unless you are very clear about the limitations of your code, people will use the library inappropriately (and I've seen some really bad implementations, even from people claiming that their code can be used for more traditional numerical work).

To me it's the same as whether your default random number generator is a CSPRNG or just a PRNG, and generally it's safer for all involved if it's the former. The latter should exist, just with lots of warnings and guidance.