Like anything electrical, it can be hard to diagnose. My '99 Sienna's power door acted up for years. You'd press the button to close it and it would close then immediately reopen. Sometimes you'd press the button multiple times before it would close. Sometime I could put it into drive and press the button, the alarm would sound but the door would close. Sometimes I had to turn off the engine and slowly pull the door shut. Eventually it quit working.

Since you're getting the latch to sometimes work by disconnecting the battery seems to me that the mechanical part is working (along with the defroster and wiper) but the ECU is having a problem with the latch mechanism. When I replaced the ECU on the Sienna the power door worked fine after that.

If you've tried oiling the mechanism and you don't see any rust or contaminants visible on the mechanism, to my way of thinking the only thing left is to replace the ECU. You could take it to the dealership and have them check it out but not replace anything. Just see if they can confirm mechanical or electrical. Maybe they will not charge more than a half hour of labor for that.

@kindpunkjames

Try to pull the rubber boot between the body and hatch and inspect the wiring there, if you know you're way around a DMM I can probably dig up the related FSM & EWD pages to test the wiring and module.

It's possible the back door issue is having intermittent issues, but it could also be that the wiring that flexes in the hinge area is breaking/broken internally and is causing intermittent connection issues. This isn't entirely uncommon on doors/hatches that see a lot of consistent use on older vehicles (usually driver's doors and passenger's doors as they get the most use).

Thanks for the suggestions everyone. I went ahead and bought a new lock assembly and it's working!

My main guess is that the motor in the assembly died from rust (have had an issue with water getting in, also ordered a replacement weather seal for the back window).