Proof of Work matters in the generation of new coins *within* the Bitcoin sysyem (there is no way to generate new coins without doing work) and the contribution of this work is what secures the blockchain and the security of the system.
The work done to generate the coins *does not transfer* outside the system. It makes Bitcoin hard money, yes, and gives it value, but value is not Proof of Work. Proof of Work is an approach *within* a system to make ppl expend computational effort to be able to do something. Ppl spending satoshis to have a relay pass along a message are not doing proof of work, *nor* is it a proxy for PoW (it doesn't transfer), they are simply spending something of value. They are making a payment. They could spend something else instead that can be transacted digitally such as stablecoins or shitcoins (while they still have value). I understand and aporeciate why ppl might not want to integrate either of these into a system and I would sympathise with their thinking but from the technical point of view, any exchange of value is all that is needed. Bitcoin serves because it has value, not because it is a proxy for PoW. That is a nonsense.
Understanding PoW is fundamental to understanding Bitcoin, in a technical sense. We see a lot of nonsense on Bitcoin social media these days with ppl talking about their own personal 'proof of work' when they've worked to do something. Yes, work can be virtuous but real world work and computer science work (computational effort) are two quite different things. The work chosen for the Bitcoin system is set to be deliberately non useful - it doesn't solve any problem useful to the real world.
Perhaps I'm over explaining now, but I remain surprised at yr misunderstanding of how PoW works. It only works within the system in which the rules for what work matters are made. This PoW then, in Bitcoin's case, creates value for the currency generated. For nostr to utilise PoW, clients would have to expend computational effort "within* nostr. They can't use Bitcoin as a PoW proxy because that is a nonsense; literally it makes no sense in computer science terms. You cannot have a 'proxy' for PoW; you must simply do the work required by the system.
Bitcoins do not represent PoW, they represent value generated from a system which, for sound technical purposes, utilises PoW internally.