So, for a normal case where multiple people want to trade with multiple other people/merchants we would need a centralized node with connections to every single one of the clients and every single one of the merchants (so a "bank"). Both clients and merchants need to open a channel on L1 (so it's goddamn slow or expensive) to that node and freeze in the channel the full amount of tokens they would wish to spend with every single peer or merchant in the future (so basically a "deposit" of all or at least a lot of the funds in a "bank"). And every merchant would need to do that too for the full amount of anticipated transactions. And then this system will work on L2 somehow, if the system will correctly calculate paths and and sums in all the channels there are. To exit the system, channels must be closed on L1 (slowass or expensive) and if there are not enough funds in the channels, create new channels with more funds, again on L1.

Even in the most absurdly centralized scenario, the system is practically unusable as soon as there are more than a handful users. Or people just stop pretending and use custodian tokens, aka bullshit IOUs.