Oh that's true, there's a subtlety: I was remembering the fact that you can't prove that you paid, and not someone else, (because others on the route have a preimage), but here that's irrelevant, because arbitration will only require proof that the payment is made, not that a specific person did so.

Hopefully we get more sophisticated in future, though I didn't expect it yet: there is no good reason not to use on/offchain swap markets for this kind of thing (they kinda already exist, but I don't think any are p2p(?)).