If everyone who sends bitcoin, would choose minimum fee (maybe even a zero) how would that change things?
That's probably true. People think they can transact any amount of money for 'free' using Bitcoin, which is completely wrong.
When the fees will get extremely high (in $ or € or any other fiat currency) more of people will decide to mine
That's not true. When fees gets higher, people will turn their backs on Bitcoin (except investors and enthusiasts), and proceed with online payment processors like PayPal.
which means there will be more Hash power, which then leads to faster transactions.
Not everyone can afford mining gears, let alone electricity and climate conditions. One needs a Massively huge Hash Power to be capable of processing/confirming transactions faster.
Maybe this high fees and long transactions times are really work of men (spam transactions).
Weird, but true.
And who would benefit from this the most? I would say miners...
Of course, they are. That's why there are a few persons who can manipulate bitcoin (or at least change its directions) because they own a huge chunk of Hash Power (someone like Jihan Wu), and respectively they get a massive amount of money.