A flat electrical hotplate, as presently found on many electric cookers. This one has a central pivot mechanism. The hotplate is like a flat plate, so appears as hotplates currently appear. In the middle of the underside is a sort of blunt point that acts as a pivot. Toward the outside edge of the underside is a trio of actuators.
These actuators can operate at hotplate temperatures, and they only have a travel of a few millimetres, say, 7mm at the most. Their travel is linearly and smoothly controllable across the whole range of travel (quite slowly over such a short travel). These three actuators would normally sit with equal normalised protrusion, giving a flat and levelled hotplate (even if the cooker itself isn't).
An electronic three-phase sequencer controls the protrusion of each actuator to cause the hotplate to pitch, yaw and tilt, resulting in any pan or pot cooking on said hotplate to be gently stirred or swirled to some small but possibly useful degree.
Operate a 'stir' control beside the temperature control for that hotplate and the actuators start to gently take it in synchronous turns to pulse in and out, with appropriate timing, to cause the hotplate to gently tilt around in a slow but controlled, continuous slight wobble.
It would have to be built with strength and safety in mind, and if one fails, they should all deactivate and return to neutral protrusion. It might find a home in the home, or it might, due to economics, only prove worthwhile in the more commercial-scale cooking operations.
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I want this so my hotplates are level - however, the reason they're not is because it's a cheapo cooker! I suspect that ones expensive enough to have this new feature would already have flat hotplates! (I'm not convinced by the stirring thing :-(
These gimbal mechanisms already exist for cooking on sailboats and ships to protect the galley slaves. You can procure a barbecue on self-leveling gimbals from your boating supply store. The mechanism would work on almost anything.
Rods, I'm not sure you could create enough motion in the liquid to have a "stirring" affect but if your purpose is merely to prevent scorching I think it would work.
Regarding stirring: In one recipe I found - Marbles - as an ingredient to improve self steering
i think a self stirring hotplate is a good idea, but not by your method, i think someone needs to start producing chemistry lab hotplates for the market, they have a rotating magnet beneath the heating element, and another magnet coated in plastic is dropped in the pot, when the lower magnet turns, it induces the same rotation in the other magnet.