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Right, the key point here is that, with a few exceptions like the one-time pad, there have NEVER been proofs that ANY of the cryptosystems we use in practice are secure! They all depend on unproven conjectures about computational hardness -- at the very least, the belief that P!=NP.
You could argue that the currently deployed systems have been "battle-tested" for longer than the new quantum-resistant ones. In reality, though, problems like factoring and discrete log have been battle tested for ~50 years, whereas lattice problems have been battle tested for ~25 years, so it's not even that huge of a difference anymore!
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I've heard from someone working in cybersecurity that for post quantum cryptography, there is a problem, that there are no proofs for the algorithms. It was mentioned as sort of a "gotcha", in a conversation about their enthusiastic 7 year q-day timeline. I was under the impression that there were no proofs for current cryptography standards either, hence algorithms like SHA1 being exploited. Are there significant differences in our confidence, between pre and post quantum cryptography, other than how battle tested they are?