Bitcoin miners affect the difficulty level directly, by mining or not.
Furthermore, there are a finite number of bitcoins to be mined. The higher the difficulty, the faster they are being mined. Therefore, the scarcity of each individual coin is going UP as they are mined. Let me say that another way: before a coin is mined, it is still available for anyone to get by mining. After it is mined, it is in a private wallet number, worth whatever the market says the owner should reasonably receive for it. As coins are mined, the supply of total available coins goes DOWN. As long as demand continues, price will go UP. BTC become more scarce with each coin mined, because there is one fewer to mine in the future, and every one to be mined after it will need to be done so at the new (assumedly higher) mining cost.
The difficulty level might be thought of as a direct index of the COST of mining a new coin. As that cost increases, supply for coins goes DOWN. Ceterus paribus, supply DECREASES, price INCREASES.
You are wrong. Coins are mined at a mostly fixed rate regardless of difficulty.
> BTC become more scarce with each coin mined
They are actually becoming less scarce since the total amount of coins in circulation increases every ~10 minutes (until there are ~21M coins in circulation). Believe it or not, supply is increasing but demand is increasing at a much faster rate, hence the price going up.
Mining difficulty on the other hand has pretty much no influence on price (although price does have an effect on mining difficulty).
Scarcity of coins to the market goes down w/ each new coin mined, you're right. BUT, because we know exactly how many coins there are available to mine, doesn't this make the total number of coins available to you via mining go DOWN? If all the world's gold was in a single, perfectly measurable and publically accessible vein, would someone moving it from that publically minable place to a private reserve make its scarcity go UP or DOWN? There is the risk of the blockchain failing drastically, but past that it can be assumed that you have THAT MANY FEWER COINS available to mine, in total, right?
I may be thinking about this in the wrong way entirely, appreciate the comment.
Miners invest in mining in expectation of price rise. They could equally go to exchange and buy bitcoins in expectation of higher price in the future. Whether they are correct or not will be decided not by them, but by all other people who demand bitcoins and bid up the prices.
If there is no new demand for bitcoins, price will not grow because there won't be competition for buying them.
So actual users of bitcoin-the-currency don't really care what the price of a bitcoin is, as long as it is stable enough for them to complete a transaction. They don't need to buy 1 or 0.1 or 10 bitcoins, they need to buy exactly $100 or $10 or $1000 of bitcoins to complete their transaction, and the actual price per bitcoin doesn't matter.
In other words, growing price of a good in a fixed supply sends a signal of its bigger liquidity. Meaning it's even better medium of exchange for you to accept (even if you don't invest in it yourself).
Things don't need to be priced in Bitcoin for it to have value and to be used in exchange with this value. Things will be priced in the most stable, most liquid commodity and today it is fiat currency. Bitcoin needs to be spread to many more hands than it is today. This is what happens today and we are still at less than 0.1% of global population that is even aware that this technology exists.
I think most money is created by private banks, not central banks.