This is the chart of Bitcoin's Avg. Transaction fee:
https://bitinfocharts.com/comparison/bitcoin-transactionfees.htmlRight now the median fee is about $2. After lots of halvings the reward won't be enough to satisfy the miners. I guess that after a decade, bitcoin will be more widespread, as a result, more transactions will occur every second. Won't this increase the median fee? If it does, bitcoin will start being a not-so medium of exchange. Imagine paying $5 for a transaction. Even paypal doesn't charge that much.
I do not think that the price of bitcoin will become higher than what it is now. Just like you said, bitcoin will become more widespread in the next decade and will result in more transactions made every second, then it means that there is more money to be made for the miners.
It is just simple mathematics. If there are 10,000 transactions made now at $2 fee per transaction, then there is $20000 transaction fee made. If bitcoin becomes widespread over the decade and there are 100,000 transactions at the same $2, then there's $200,000 made.
There won't be a need to increase the transaction fee.