Yeah, mixers can be useful to bring cooler liquid down to the bottom. Bigger the delta T, the better the heat transfer and efficiency. We used to run direct fire, without agitation and it was slow. At least in the brandy world, it is believed that the high heat and copper contact in a slow warm up contributes positively to flavor. Not sure if that is the same for rum or not. We switched to direct steam coils, still no agitation on that still and warm-up is faster, but still slow, agitation would help with that, but it is fast enough, and the flavor profile seems unaffected. We used to pre-heat our wine for the still to 120 to shorten warm-up. Might be an option for you if you have an insulated tank, drain the still through a coil in an insulated tank (or use another heat exchanger of your choice) to save the heat for the next day. My understanding is LP is typically much more expensive than NG, (although I'm sure that varies depending on location), so that could be another reason to try to save/reclaim heat.