I've seen some people use a magnetic mixer/stirrer. Several different plans out there, many involving a rare earth magnet glued to a cheap pc case fan (for rotational motor) then using a stir bar inside the container. They have coated stir bars that won't nick/break the glass, they use them in glass beakers all the time.
This one they're using to mix shellacs and varnishes but people have used the same basic method to stir their ejuice. It would be a lot safer than a ball bearing inside glass bottles. Some people just shake and let their juice rest. I've seen equipment in professional mixing labs that use devices which are a combination of mixers with heating elements underneath to control temps and mix juice batches.
No evidence that the way most people do it at home is any worse off but there has to be some reasoning behind it or labs wouldn't go to the trouble. Thermal controlled mixing vats are likely much more expensive than paint can shakers found at home improvement stores, if shaking were just as effective I'd think they'd go that route. Unless they're speeding up the process to get batches out the door or something.
Personally I've still been less than stellar at diy juice making though may look into building one of those stir machines like in the vid. If nothing else because I'm lazy and beats the hell out of vigorous shaking
I saw one vid of a guy who strapped his bottle to a reciprocating saw. I have one so that would definitely be an option but it shook it so much it turned his juice to muddy looking froth. Maybe overdid it with too much of a good thing lol.