You can - look for a hot chocolate mixing mug or magnetic mixing mug on {large retail site}. Gave my son one for Christmas a few years ago cuz he's a nerd like me and adores hot chocolate. He loves the thing!

Yeah, I think most "solutions" in the kitchen are far too concentrated. I wouldn't expect a stirring bar to do much for stirring dough or any but the thinnest batters.

Homebrewers use them, for yeast culturing or making starters. I guess that's fairly lab-like, but they do it in kitchens.

Yup. Anything bigger than 500mL or anything more vicious than a water solution gets an overhead stirred in the lab.

Magnetic stirrers are great for small scale or even larger scale if it’s just mixing water with acid (for example), but they stop working well after that.