The Vietnam government has banned rooted phones from using any banking app

The Vietnamese government has mandated all banking apps to detect if either the phone has been rooted, the bootloader has been unlocked, or ADB is enabled and force quit if that's the case.

Do those same banks have websites that you can access from a computer with root access? Most likely, yes.

Smart phones are not personal computers. They're shopping/government/etc terminals. You don't and never have controlled them, even with root (re: tight integration of the baseband computer which only the telco has a license for, not you). Their best use re: computing is acting as wifi hotspot for their cell telco CNAT connection. The time to stop using them as computers is now, not when your local government passes these laws. Apple is already forcing it and Google has shown it's cards even if walked it back temporarily.

Eventually though I suspect that web access to banks will be rescinded too, much like HMRC in the UK no longer permits companies to submit their taxes through the websites.

In the future, everything will need an 'app'.

Isn't that what happens in Europe with most rooted phones and banks too? At least I can remember my banking apps stopped working.