We have already seen transactions as high as those high peak days, we have seen the second biggest day in bitcoin transaction history, nothing happened at all. Which means people are already buying bitcoin like crazy and they are sending/receiving bitcoin like crazy without any problems at all.

Sometimes it changes the time period which makes transactions take a bit longer than usual but the fee doesn't change at all. Thanks to developments in stuff like segwit and then lightning network the transactions are now cheaper and they are even faster as well which makes it super easy to do all of this. So no, even if bitcoin goes to a million dollars each (lol I wish) there won't be a problem because we do not have a static blockchain, we have a dynamic blockchain that runs equally no matter how small or big it gets.