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.
This seems plausible but do miners really drive the price of bitcoin as much as you say they do?
Well, we can't say this directly but yes, it mostly depends on them because there are some big mining companies like BitFury which built largest datacenter in my country 1-2 years ago, they invested millions here. If mining won't be profitable, than there is a need of bitcoin's price rise because without good price, it don't worths and big players needs to try something to solve that problem, also it's sometimes good for exchangers.