That nightly elevation change...

I guess that is the nightly altimeter auto calibration, which uses the phone to get your location and then set the altitude for that location based on the watch's DEM map.

It's described on the following page under Auto Calibration Nightly (that page is for Fenix/Epix, but should apply to 965 as well): support.garmin.com/.../

Mine has done this since day one. I don't have the phone connected apart from the occasional weekly sync but every single night it will put me consistently 150ft higher than I am despite manually or dem setting the altitude before bed. Then every morning once I put the watch on it'll drop 200ft in a few mins of walking down one flight of stairs. I get weird jumps in my barometer most days, vertical 10mb ups and downs. I can spend all day outdoors in the garden and it will show me climbing & descending 100s of feet an hour. I set the altitude & barometer occasionally but an hr or so later it's 100s of feet out and all over the place. 

If you have manually calibrated the altitude in your home location, the watch should remember that altitude when it next calibrates in the same location.

If you don't have s phone connected at night, it shouldn't affect the calibration.

OK, I'll start by shutting the phone off for a night and see...

Yeah...this doesn't happen for me. I can manually calibrate and it'll drift off, then I'll either manually do it again or wait for a gps fix and do dem but it's quicker just to enter manually. I don't think they 'remember' your home location or alt it just gets a gps fix then reads it from the dem file.

Yes, correct. After an activity I get the correct value, right, but that gets messed up every night at 02:30. 
And this is at home, where most of my activities starts an ends, and where I sleep. One would think that the watch would eventually notice this, but no…

Strange. For me it works, my altitude is typically slightly off at the evening, and nightly calibration fixes it to the correct value.

(But I keep the phone connection on all the time.)

So it is usually just 10meters of calibration?  That seems pretty minimal , keep in mind that temperature effects that sensor (hence when you pause for a bit mid ride/run outside in very warm or cold conditions (see temperature change on graph of activity) often elevation will also spike up/down. Just looking at a 35km run I did with a group with a couple long breaks (winter 0°C) and every time we went back outside there was a ~12m/30-37foot drop instantly before slowly coming back to 'normal'.  So overnight calibration while under blankets or in a cold bedroom etc... can tinker with it .  Especially if on a different floor of home/apartment.  really 10m / 30ft... pretty much in the realm of accuracy that I would expect.  

Looked at the starting elevation and final elevation of runs from my house.... all are close (feet) but def NOT the same, but well within what I consider "very good" (1meter)

913/920

900/915

909/919

897/913

900/919

seems like looking at it that the higher and lower elevation readings coincide a bit with cooler and warmer runs.  Although those warmer runs also are ones I had been awake for an hour or two around the house before heading out.

It varies. A week ago it read 400+ meters, I live at 357