So - and thank you for bearing with me on this - individual BTC don't have a unique identifier that allow one to trace its transaction history? I must admit, I always assumed that they were uniquely identifiable.
Yes. In the first place, BTC are divisible to eight decimal places, so there would be a lot of these IDs if they were to exist. And there's no reason the divisibility can't change in the future, so these IDs would have to be quite complicated.
A (non-coinbase) transaction is valid if its inputs equal its outputs, and if its inputs come from the outputs of other transactions, which in turn are valid if their inputs come from other transactions, all the way to one or more coinbase transactions. This is sufficient to demonstrate that the transaction is valid - ie it's using coins that exist instead of creating them from thin air. That is all that the protocol cares about.
