Can we have a strictly monotonically increasing/decreasing sequence generated by a distributed system?
You're right that this is an impossible problem to solve in an asynchronous distributed system, and you're also right that it would solve a lot of problems if we could get a totally ordered clock. But it only solves "all" our problems if the clock has the additional constraint of a meaningful relationship with real time.

The two best solutions we have are partial ordering (Lamport clocks) and consensus (which can be fault tolerant but is typically driven by an elected leader). The latter provides total order and works well in practice (with randomized timers), but is impossible to guarantee in theory (FLP).