While absolutely true, I think (from experience) that people underestimate both, how often you end up not being hired (even if you make it to the final round),

While absolutely true, I think (from experience) that people underestimate both, how often you end up not being hired (even if you make it to the final round), and how being stuck back with the old company with this out in the open, is not a pleasant thing. Not every company will blatantly call your former/current employer for a reference, there's procedures for that, and you can certainly control this a bit, as an applicant. My suggestion would be to openly make this a topic of discussion with your potential new employer. Tell them that you'll have to manage the time as you're still employed and can't just run off during the day easily. They'll understand, even the big ones. I've been able to arrange all interviews at a time I found convenient, and that includes the big US tech giants.