That doesn't answer my question. If it counted up then it would be up to each hop to set its own policy. Things wouldn't loop endlessly in that scenario either.

Then random internet routers could break internet traffic by setting it really low and the user could not do a thing about it. They technically still can by discarding all traffic whose value is less than some value, but they don’t. The idea that they should set their own policy could fundamentally break network traffic flows if it ever became practiced.

This is a wild guess but: I am under the impression that the early internet was built somewhat naively so I guess that the sender sets it because they know best how long it stays relevant for/when it makes sense to restart or fail rather than wait.

It does make traceroute, where each packet is fired with one more available step than the last, feasible, whereas 'up' wouldn't. Of course, then we'd just start with max-hops and walk the number down I suppose. I still expect it would be inconvenient during debugging for various devices to have various ceilings.