I suppose, but wouldn't it be better if cash simply didn't have serial numbers?

Sure, but it's moot as you said

Oh I didn't think about this... A bank can't force you to "deposit at an ATM". A counterparty can force close your channel. So even lightning is jot as good as cash. Also cash has no payment processors. Lightning doesn't have to have them, but in practicality almost always does, in the form of large liquidity providers offering channels as a service.