I think the mining difficulty (while not an exact match to exchange rate) is a strong and correlated signal of exchange rate. In addition it is easily obtainable / calculated for all miners in the system.
Actually, you are right. I retract some of my smugness from before.
Mining difficulty is related to total hashing calculations / time. So mining difficulty is also related to fiat through hashing efficiency * electrical cost.
It is hazy because average hashing efficiency is an unknown variable. Over time it converges to the most electrically efficient miners. However, its value is further obfuscated because the "future value" of bitcoins is considered unbounded.
That leads people to consider wasting "dollars" today by spending more than a coin is worth. While hoping to make that up over the long term via appreciation.
Stable money removes some of the unknowns and makes solving that calculation MUCH easier.