Yes it recirculates. I have an ANOVA one (they're under £100 nowadays I think) which keeps my insulated 30l pot (camping mat) within a degree. I give it an occasional stir. You would need to use a bag for your grain (brew in a bag). I've used mine for 100 + brews and works for me.The stick does the recirculation? That's ace. So I can leave it in my mash and it will keep it at my target temperature?
They're not cheap, but actually could be a really good choice. Have you tried step-mashes with it?