Of course you can "spend" debt, but only if the debtor is very reputable. Consider the old example: I ask you to fix my car. I don't have any money on me to repay you, so I give you an I.O.U.. You go get a haircut, but don't have any money on you either. The hairdresser knows I'm a standup guy so he takes my I.O.U. as payment instead. Later he comes to me to collect, I repay him and we rip up the I.O.U.. See how it can be spent like money (we could of course add any number of people in between who trust me where my I.O.U. changes hands)?

Part of the agreement with the bank is that they guarantee (to a reasonable degree, as the FDIC puts it) to be available for collection in cash at any time. That of course makes them an extremely reliable debtor, and therefore their I.O.U.s (a.k.a. the money in your account) are virtually globally accepted as payment (not least because of the government heavily regulating the matter). See the parallels?

Also, I still would like to know what the legal nature of a bank account is if not debt. I think I've ruled out Bailment, Trust, and Agency. What else is it?

Going on a tangent here, I think what cannot be understated is the power dynamic intrinsic in debt agreements. Usually, the creditor gains a considerable amount of power over the debtor, especially if the latter fails to repay his debt (the threat is foreclosure, imprisonment, etc.). It may be difficult to see a bank account as a debtor/creditor relation, precisely because this power gradient is inverted. The bank is the debtor, but somehow they retain all the power in the relationship.

Consider what happens if they cannot pay up (during a bank run for example): it is not the bank and the bankers that are under physical threat, but its creditors (the account holders), because obviously without money they cannot survive.