No, a user does not have to broadcast an L1 transaction for any of this to work, although they will make and sign a transaction that looks awfully like one (BIP322 signed messages). That's the beauty of all this - users are completely insulated from L1 transaction fees.
I think the main question by BrotherCreamy can be paraphrased as:
In Lightning, your 'security' is ensured by your ability to 'fall back' / settle channel balances on-chain. If there is any issue, you publish a channel close transaction to the blockchain and get your funds back out.
The question is how this channel closing (and opening) mechanism is eliminated by Settlement Pools.
You're saying that you basically never
need to close your channel / pool?