The difference between gold mining and difficulty adjusted block reward is that in gold mining, you know how much gold you will mine. But as the difficulty changes all the time, you do not know how much coin you will mine.
My proposal would be more like having all gold miners work in a single, shared mine, where everybody is allowed to grab what he can. You know in advance how much gold there is and how difficult it is to get that gold, you just aren't sure you'll be the first to get it.
If the road to depletion of the to-be-mined amount is slow, then the reward (expressed in amounts of the cryptocurrency) for a given amount of PoW is relatively constant. In fact, for some kinds of PoW/reward curves it would be completely constant for most of the time, just like Bitcoin's block reward is constant most of the time (except when it halves). You know your own hashing power, so you know how much reward you can get with that.
An open question is: given that Moore's law exponentially increases mining power (different from physical mining!), is there a PoW/reward curve that allows mining to happen "slowly"? This issue does not exist in Bitcoin, since it completely compensates Moore's law (or any other development in hashing power) with its difficulty adjusting algorithm.
Your premise is completely false. The large price fluctuations in BTC are almost exclusively demand driven (i.e., speculation) and have very little to do with supply. And in any case, the supply is not fixed with respect to time the way you think it is. Obviously, were prices to rise substantially, some people currently holding BTC would sell, and the reverse is also true.
It's true that mining output is just a part of supply. If it's only a small part, then the corresponding effect would be small too.