That analogy sure stopped and made me think. What a sticky issue. SHOULD browsers do that? What if your bank website refused to allow you to sign in from a machine running unpatched XP?
I think right now there might not be enough exploits targeting banking on XP machines to justify that inconvenience, but it seems like a responsable argument could be made for both cases.
It might seem like a good idea on the surface, but think about the impact this might have on people who can least afford it (consider why people might be using old phones or old operating systems, and what impact lack of access to their bank could have on them for even a day if it's the wrong day), in the name of protecting them from something theoretical.
I think right now there might not be enough exploits targeting banking on XP machines to justify that inconvenience, but it seems like a responsable argument could be made for both cases.