I think SHRINCS could be an excellent option for Tadge Dryja's proposed recovery mechanism [1] or other recovery schemes like [2].
For now (and probably several years), people can use P2WPKH as normal - as long as the public key is not known, i.e. you don't re-use the address (or publish your key somewhere accidentally), your funds are not at risk even with advanced quantum computers.
Now if QCs become powerful enough, Dryja proposed roughly the following scheme:
1) confirm a transaction with a proof that you know the private key (e.g. a transaction ID) on the blockchain
2) reference this proof in a second transaction where you effectively move the coins.
3) transactions which move ECDSA coins that don't contain this proof are invalid after a deadline (which requires a soft fork).
For step 1 and 2, a quantum safe mehtod is needed, but this wallet is no necessarily meant to be a permanent storage, it can be an intermediate or short time wallet. So the "stateful" problem isn't really relevant because ideally you would use it only once (or a few times), that problem appears mostly when you want to use the wallet for "daily" usage, i.e. to transact every now and then. But in Tadge Dryja's mechanism, it is needed only for an "emergency".
Implementing SHRINCS on the Bitcoin protocol in the mid term (e.g. before 2028 or so) would give us the option to use Dryja's recovery method as soon as the problem materializes. This means: If really this Google study is correct and in 2029 a quantum threat is already realistic, then we have already a safety net and Bitcoin could be continued to be used without major disruptions. But the developers have then more time to develop a better solution for the 2030s.
Of course this does not protect re-used addresses and/or P2PK and similar vulnerable coins.
[1] see mailing list post:
https://mailing-list.bitcoindevs.xyz/bitcoindev/cc2f8908-f6fa-45aa-93d7-6f926f9ba627n@googlegroups.com/[2] Bitcointalk thread:
https://bitcointalk.org/index.php?topic=5571348.0