If I remove my drive from a dead computer and put in a spare one, it should boot up in the same state, including cookies in the browser. With a desktop computer and SSDs that could easily happen within the banking timeout. With Linux it is trivial to do as well.
Wait, so your use case here is that you login to online banking and while you are paying your bills or whatever your computer dies, you pull the drive from the computer that just died and put it into the new computer, boot it back up all within 10 minutes, and then expect to still be logged-in? That seems exceptionally unusual, and logging into one account seems a small inconvenience compared to replacing your entire computer. tbh I'd be amazed if it even works now. Does Linux restore the complete memory state of a dead computer when you install the drive in a new machine?
Bank logins use session cookies that are cleared when the browser closes. Unless RAM is preserved you'll need to re-open your browser, so they'll be lost.
Personally I have more use for protection against session theft than I do for moving a drive to another computer and continuing to use the same online banking session within 10 minutes. I suspect most people are in the same category.
Look at the situation with mobile phones. Half my apps on Android are impossible to back up and restore on another phone.
If my phone is damaged, "logging in" again to my bank's mandatory app means I need to fly half way across the world and visit a branch in person with my new phone.
I don't want anything like that happening to desktop devices, regardless of how small the initial steps in that direction are.
