An exchange isn't a bank, and, in theory, isn't subject to a "run".
Banks are typically "fractional reserve" institutions. The money deposited by the customers doesn't all reside within the bank; some fraction of it is loaned out or otherwise invested, so that the bank can earn profits and pass some of it back to depositors as interest. So if all the depositors want to withdraw their money, it isn't there, and some will be left out in the cold (unless there is deposit insurance to make them whole).
Exchanges don't work like that. Conceptually, all funds deposited by customers, both fiat and cryptocurrency, stays within the exchange. The exchange doesn't attempt to make profits by investing or loaning out those funds, and they don't pay interest on their customers' deposits. Instead, they make money by charging commissions on the trades which their customers make with each other. So in principle, if all the customers of an exchange want to withdraw simultaneously, there's no problem; the money is right there and they can all have what is theirs.
Of course, the exchange might encounter some technical issues processing such a high volume of transactions in a timely manner. Also, in practice the exchange probably doesn't keep all the fiat money on site; they deposit it in their own bank account, where it stays (although their bank may in turn lend out some portion, as banks normally do). So if the bank doesn't have many customers other than the exchange, then when the exchange tries to withdraw all their funds to pay out to their customers, it may effectively be a run on the exchange's bank. But this is unlikely if the exchange chooses a large enough bank to work with, or has insurance.
It's also possible that any given exchange is cheating and not actually behaving like a theoretical exchange should; there aren't too many laws that would ensure that they are doing so. But conceptually, there is a fundamental difference in the ways that exchanges and banks operate, and the former aren't subject to this kind of problem if they operate as they should.
You might be conflating this with a different issue. "The price" of a coin is usually understood to be the price at which people are trading the coin right now. It doesn't imply that an unlimited amount of coin can be bought or sold at that price, and nobody should assume that it does. If "the price" hits one billion dollars and "everyone" tries to sell, there probably won't be buyers for all of the coin at that price. The price will fall, probably a lot, until it reaches a level where there actually are buyers and sellers. This is normal supply and demand stuff, same as for any other asset.