Why not use Firefox multi account containers[1] for this use case?
Having one banking container or even one per bank would have the same effect, without the annoyances of private mode?
I tried it out, but it was much more structured than I'd like. I don't really want to have pre-defined a bunch of containers for each site that I want compartmentalized. I really like the ephemeral nature of private browsing where a single click gives me a fresh new container that I can use for whatever I want. Plus, when giving recommendations to family members, private browsing is more user-facing feature, and easier for them to use.