Putting Bitcoin as a safe investment along with gold and silver is laughable. Things that routinely fluctuate 10% or 20% in a day are not safe havens to store wealth. Gold has thousands of years of history as a (relatively) stable store of value, but bitcoin has barely a decade and it's rife with horrific crashes. There's no way you can make a credible argument that something that volatile can be a safe haven.
Agreed, but I do think Bitcoin's monetary policy lends itself to being a store of value. It just can't be done overnight. This volatile price discovery is theoretically just part of the early adopter phase, for the same reason startup penny stocks can be so volatile. Investors are speculating on a rather binary situation: mass adoption (or capturing major market share)......or failure.
If the first scenario plays out, I could see price plateauing an order of magnitude or two higher than now, with volatility gradually declining as the speculative nature of Bitcoin becomes fully priced in over time.
Gold is still considered a rather volatile asset, easily fluctuating 1-5% in a day (it fell 6% in a day last month). I could see BTC eventually occupying a similar store of value niche and degree of volatility to gold, or certainly silver. It might take decades to get there, and maybe I won't live to see it.
In theory, the artificial scarcity is should help rather than hurt the price, but there are a couple of caveats to consider. For one, plenty of things are scarce and not valuable, so scarcity itself doesn't equal an ability to be valuable or hold value. Value has to come from somewhere else, like utility. On that front, there are plenty of cryptos that are more useful than Bitcoin (faster, cheaper, etc.). Pretty much the only thing Bitcoin has going for it is first mover advantage and inertia. I don't see Bitcoin ever reaching the kind of mass adoption you're envisioning because on just about every functional utility criteria you can measure it against, there's another crypto that does it better. For these reasons, I never see Bitcoin entering a phase of price stability necessary for it to be a long term store of value.