Are you making pour over with boiling water?

I heard somewhere— maybe from Hoffman— that you want the temperature to drop over the brew to reduce the risk of extracting bad-tasting compounds. Subjectively, I think my pour over tastes better if I don’t put the kettle back on (the base, in my case).

Boiling water on a stove? What year is this?

Try it both ways. Bet that you either won't notice a difference or that you might slightly prefer leaving it off. Brewing at near boiling the entire brew might be tolerated by some pretty light roasts, but in general a little less extraction of the late grinds is typically preferable. It feels a bit less fussy to not have to keep popping it back on the stove too.