Relying on a single layer of security is notoriously unreliable. Remember good ol' Spectre? Tests have shown that the exploit could even be perpetrated via Javascript running in a browser. And if you understand what Spectre is, it's a way to infer the contents of the host system's memory, even to the point of being able to see inside another VM running on the same host.If a website has code that's trying to sniff out data, and your browser allows that, it's a browser problem right at the start, not an OS memory integrity thing.
Ideally, one wouldn't even use the same physical machine for online banking/investing/shopping and gaming. However, running them in separate VMs, or using features like VBI, should make Spectre-class exploits more difficult to pull off.