So, as I see it, every 4 years btc is in danger of becoming not used and mined, and the way out of the potential catastrophy is the price growth.
My question is how does it happen that the price actually grows and mining becomes profitable after halving? How can people make the price grow to an exactly needed amount without people actually trying to do something to save btc? Why does it work?
Not used and not mined are two different things.
The bitcoin that has been mined can always be used so long as someone knows the private key to enable transactions from the wallet where the bitcoins are located.
Mining technically could stop at some point, but it probably won't until we've reached the end of the supply to be mined. And even then "mining" will occur to verify transactions but rather than producing new blocks of bitcoins these miners will be getting a fees for the confirmations they're producing to prove transactions are valid. Some people consider this still mining, because the miners are doing the same work BUT they aren't discovering or minting new coins, they aren't adding to the supply base so in my opinion they aren't mining. At that point they'll be validating transactions.
As time goes, hardware and software is developing and there is no reason for panic. Also as you see now, price rised and it's between 900-1000 usd and if mining won't be profitable, price will continue to rise, it's like must to happen process.
This seems plausible but do miners really drive the price of bitcoin as much as you say they do?