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.
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.