Bank run theoretically possible, but it is very unlikely.
As we have interbanking borrowing mechanism and even swapping facilities between countries central banks nowadays for any cash needed instantly.

While central bank (that has the ability to print money) is the last lending resort, aka bank can borrow from central bank if really desperate time.

People withdraw (bank run) from Bank A, when it doesn't have enough cash money, it will request Bank B to borrow them to facilitate the "bank run". By then those people have the cash money from bank A, will go to bank B to deposit back. (so pocket A to B only), unless everyone withdraw billion and keep under pillow, then yes, a real bank run may occur that collapse the bank.

It is better to worry on "personal financial bank run" aka run out of cash to survive instead of bank run.

The reason why investment like unit trust, or equities are not suit to be emergency fund, because, at the point of emergency, those investment may be making a loss. Unit trust is not guaranteed to be making money one.
There are fairly a number of unit trust out there are still making a loss despite invested for 3,5 years or even longer.