"because it gives you control of the funds without third-party services."

How?

"you can open, close, and manage channels. final settlement occurs onchain so you keep your funds through the process."

None of this is explaining how you can hold the funds without third-party services

"requirement of an underlying network has no bearing on self-custody"

Did you misread "self transacting" as "self custody" or are you still just repeating your view without corroborating it? Where is the explanation of HOW requirement of an underlying network has no bearing on self-custody?