Imagine people had refused to use the calculus because no one could really answer what an infinitesimal value really was.
Sometimes you just need to make progress.
My understanding is that it was rejected by many mathematicians for precisely that reason, but that many others took up the challenge to answer the question rigorously, resulting in some major leaps in theory. On the other hand, it's good that many others were able to overlook the issue of the infinitesimals and put the techniques to lots of practical use. The lesson I take from this is that both first principles questioning and suspension of disbelief can be valuable tools for learning and discovery.
I don't think the GP (Kluny) or GGP (OscarCunningham) disagree with you on that point. If I understand them correctly, it's fine to have unresolved mysteries in your understanding, so long as you are honest about it and explicitly label it[1]. In this case, "hey, we can't quite put infinitesimals on a solid grounding (or Dirac Delta, or whatever), but it doesn't break anything else, so let's see if we can figure it out some other time."
There's a huge difference between "it's not worth our time to resolve this now" and "non-rigorous thought is okay, reality doesn't have to make sense, just do what the high-status people say, they're smarter".
[1] There's a practice in some groups to explicitly say "it works by magic" to avoid false understandings and to flag unresolved issues.
Similar logic puts some people off of Christianity - x is in the Bible, and we know the Bible is true because the Bible says it is true, therefore x is also true. Hmm. Someone who tries to sell you that logic is also either lying or wrong, now you can't trust either that person or the Bible.