I disagree. Avoiding PKI and post-quantum security correlate very much. Even under plausibly post-quantum assumptions we only have a couple of assumptions from which we can build public key encryption. In contrast, here they avoid all use of public key cryptography which makes it provably post-quantum secure. It’s not using a buzzword for the sole sake of selling the paper. In general, using “minimal cryptography” (like random oracles / one-way functions) translates to real-world efficiency because you can instantiate these from a plethora of different concrete candidates.

> Avoiding PKI and post-quantum security correlate very much. Even under plausibly post-quantum assumptions we only have a couple of assumptions from which we can build public key encryption.

These statements presuppose an overly expansive definition of PKI, i.e., distribution of keys for public-key encryption. A more conservative definition is PKI = availability of trustworthy publicly verifiable signatures (i.e., public-key certificates). Post-quantum signatures can be based on target collision-resistant hash functions, like XMSS.

The paper assumes pairwise private and authenticated channels. While in practice this is not necessarily a good substitute for PKI, in theory it is a strictly weaker setting.

I'm only somewhat into cryptography, but is PKI considered as sketchy in some way?

No.

The proofs for the security security of most of the methods of public/private key systems are weaker than "you can't reverse this hash function".

They are still robust in the face of computers that may physically exist in foreseeable futures. Elliptic Curve Encryption was adopted for being post-quantum twenty years ago and more work since then I haven't followed.

The person I am arguing with is imagining a future pessimistic beyond what most would consider reasonable.

Elliptic curve crypto is not post-quantum. (Indeed it’s likely to be broken before RSA if cryptographically relevant quantum computers occur).