> The higher the difficulty, the faster they are being mined. Therefore, the scarcity of each individual coin is going UP as they are mined.

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).