Yeah, but increasing block-size is not a longterm solution.

Are you sure? That sort of statement is hard to answer because it doesn't say what you think long term is, or how much you expect Bitcoin to grow.

Satoshi thought it was a perfectly fine long term solution because he thought hardware would get cheaper as fast or faster than Bitcoin would grow. You may disagree with him, but as we're talking about the future are you 100% certain he was wrong? I did calculations a long time ago that suggested even with today's hardware (+some software optimisations) it would be feasible to keep up with Visa.

Hardware improvements can be unintuitive. There's a spreadsheet here that lets you play with various parameters. 

https://docs.google.com/spreadsheets/d/1PJvrAAOVYVszfRRLhKqd1R9lRiOAImzAfdeb6ajATEY/edit#gid=1451669128

(note: the spreadsheet says avg txn size is 250 bytes, but if you check the formula for the middle column, it does actually use 500 bytes as the multiplier hard coded).
 
Necessary higher fees are a logical consequence of lower subsidies. Bitcoin was basically free to use at the beginning because miners got paid with new coins at  the expense of those who already hold coins. Eventually there needs to be a mechanism which matches supply and demand.

That's not clear either, I'm afraid.

Remember that there's an upper limit on how high Bitcoin fees can go. When fees become higher than what the banking system charges, many users won't use Bitcoin for moving money around anymore. Fees cannot really go much higher than that even if you assume the currency is still attractive for other reasons, because people would just sell their coins for fiat, move the fiat, and buy back the coins the other side.

The way mining will be funded in future is an open question. There are differing proposals. Still, even with a higher hard block size limit, miners can always refuse to mine transactions that don't include a particular fee. So if you're worried about this, miners aren't being forced into any particular policy.