If getting the spec right is way easier than getting the implementation right, then strict waterfall would be the most effective way to write programs, and the holy grail of software engineering would be to automate the process of generating implementations from formal specs (there are some people who think exactly that, but that has been the case for about five decades, with effectively no impact on how software is actually made.)

The reality is that we often overlook issues with the spec until we try to implement it, or try using what we have implemented, even when we are trying to get the spec right first.

I have no idea where you're going with this, but getting the spec right is very different from actually finding a proof for it. I never said actually finding the proof was always easy. But I will say this for sure: getting the spec right and finding a proof for it are jointly much easier than writing a nontrivial bug-free program without a theorem prover. The success or lack of success of waterfall is not relevant here, as that has very little to do with formal verification; additionally, the success of most software methodologies is not measured solely on whether the end product was correct.

If programmers sat around writing <10K line pure programs with no user interaction, they would have just as lovely a time as Mathematicians. If theorem proving in mathematics is easy for you, it is because your aren't trying to prove hard things.

Getting a spec right is not the same as proving a theorem. It is the same as stating a theorem. And I absolutely disagree with you that making small pure programs magically makes them bugfree.