What is the idea here? Open a channel with each website that I visit. Lockup the BTC that I will ever send them but only send it to them slowly each time I visit it.
Yes. In reality, users would have a connection open with one or a small number of routing nodes.
The end result is very cheap and quick funds transfers to any node on the network. This lightning network infrastructure would make micropayments feasible.
Granted, there are still problems to solve. But this is the dream.