However, price affects how many people choose to enter (or exit) the mining, thus changing the difficulty.
Right, but if the price is too low to justify starting new mining (meaning buying equipment to mine) and people want to transact in btc, then they will have to buy it. This will put the price up. This is why I think these three, price to enter mining, price of a btc, and difficulty, are all related by an equilibrium.