For work stuff my habit for maybe a decade or more has been to have work buy me hardware and the work hardware does work stuff, so this conflict never arises. The closest is maybe a previous employer paid me a retainer and obviously they didn't buy me a special laptop just for like one conference call a year on retainer, so I did that from my PC.

But yes, in a Foo container, all the various Google things (Docs, GMail, their Cloud offering, Youtube...) are either not logged in at all or they're logged in from some Foo context.