I did not heard about paynym yet, but I think every innovation for LN is desirable.
Maybe it is also possible to build private(like Monero and Zcash) blockchains/sidechains based up on the original chain by pegging the tokens with BTC. Such as Liquid or Binance chain. So in fact using BTC as standard for funding/backing additional transaction systems.
Paynym is not a LN innovation. It works without LN but I HOPE it will soon also work with LN. ;)
I don't think there will be private sidechains possible.. cuz if some BTCs are private, you don't know how many BTC there are exactly right now.
That's why Grin is never going to be a Store of Value. But as private medium of exchange the exact amount isnt that important as for BTC (cuz of SoV).
:]