After sitting overnight in a warm-ish (0C) garage the handle started working again. I think it has a temperature sensitivity.
Hmmm... functional handle ..sometimes. Cold sensitivity. Could be poor electrical contact of a connector, maybe the door handle harness plug on the car side if replacing handles (as many times as you've had) has not gotten to the problem yet. With plug-n-play modular replacement the module works for awhile .. because it sufficiently disturbed the connector back into working condition for a while.
It could be a poor wire crimp on a pin, a dirty pin, or a stretched open pin not providing enough contact.. It could be wire flex fracture at the crimp on the pin.
What I sometimes do in cases like this is try to get at the failing part while leaving everything in its place in the failing condition, then wiggle, flex, move, hold at funny angles, warm the connectors .. (hair dryer) see if the thing becomes suddenly operable again if you hit some sweet spot. In hot environments, I spray canned air on points to freeze them.
Take the connector apart on both sides look for cleanliness, tarnish, corrosion on any pin. Bright light, jeweler's loupe, or other strong magnification helps. Gently tug wires from the back of the connector. Did one just break and pull out with ease?
Take a properly sized mating spare male pin (meant for the connector) and insert it into each hole of the connector side with female pins, using just fingertips to feeling for loose or weak insertion / removal forces. (You can sometimes find extra unused male pin in a similar connector on the car or the same connector ... extract it as a probing donor.) Any weak holding female pin might be improved with a gentle squeeze of needle nose pliers when extracted.
Using a contact cleaner and electrical contact lube can mean the difference when putting it back together.
A non-responsive handle (won't present) is not getting the message. Power or data. Can at least test that power is getting through the connector using your VOM testing the power carrying wires with an insulation displacement piercing probe (fancy name for needle).
An intermittent wire problem might be mid-harness anywhere it is overly flexed (going into the door from the hinged pillar side) ... or any connector along the way backward to the commander and supplier of life.
Mechanics can really rack up the hours diagnosing these things. Sometimes in frustration will start cutting out connectors and hardwiring (solder and tape) each wire pair... to make sure they never see that problem again.