Let's suppose, just for argument, that the majority of the miners decided to raise the cap to 210 million BTC, and lower the difficulty so that mining became 10 times more productive in BTC terms. What would happen?
Note that the total supply of bitcoins would not immediately increase, there would still be only 12 million BTC "in existence". The flow of mined bitcoins would increase by some factor, but it would still be years before the remaining 9 million "old" BTC are mined and the "new" bitcoins start to be broken into. The market value of bitcoin may drop somewhat, because of the loss of the "scarcity" aura (which was lost anyway when the altcoins sprung up); but it may also go up, because the future viability of mining would look less uncertain. The utility of bitcoin as a medium of payment would not be affected (except perhaps momentarily by the change in market price); on the contrary, bitcoins would eventually be more evenly distributed, and a larger slice of the pie would go to those who are actually making bitcoin work. And the increased "inflation" rate due to higher mining output would discourage hoarding.
Miners who chose to remain faithful to the "classical" protocol, with its cap and difficulty, would be making 1/10 of what their colleagues are doing; and will lose all their output if they fork and the other chain prevails. Why would they do it?
You seem to be assuming that all miners are hoarders. But they already must sell a large fraction of the BTC they mine in order to pay their energy bills and other expenses. Many miners probably sell all their production, in order to have the profit in cash now rather than hope for the hypothetical future when one bitcoin may be worth a zillion dollars. These miners do not care if the present BTC owners will own only 5% of all the money in that fantastic future, instead of 50% of it.
Note, I am not saying that this will happen, that its is likely to happen, or even that it would work in exactly those terms. I am only pointing out that miners' greed (or simply their need to make ends meet) may well destroy bitcoin's supposed scarcity, rather than protect it.
Ultimately it makes no difference at all if the supply is fixed or not as long as miners get the transaction fees from mined blocks as a reward once the cap is reached. All you are really doing is changing the denomination. Proof of Work in the form that is being used will raise/lower difficulty in accordance to the computational power being thrown at the network. What difference does it make if you mine 50, 500 or 5million coins in one block or merely receive the fees?
The price will converge to what the market thinks it should be.
[edit] to clarify this. miners will mine if it is profitable to mine, even if it is just for the transaction fees