Lightning network is peer-to-peer, not as in Bitcoin, but still. Third party liquidity providers do not gain custody of the peers' cash, so it meets my conditions to be called p2p. It is just less permissionless, the less the number of your channels.
all crypto is PERMISSIONED
they all require a signature from someone..
you are confused between self custody(your own permission is needed if someone else wants to be paid(they cant take money from you))
vs cusodians where you need a middle mans permission to make payments on your behalf of balance they hold they suggest was yours