Elevation consistently inaccurate (reading too low)

Your readings seem pretty strange.  Almost all of my inaccurate elevation gain readings are under measuring rather than over.  Your spikes seem like something that might happen with a noisy GPS-based elevation reading.  I assume you don't have "elevation correction" active in GC and/or Strava. 

I have seen drastic improvement in elevation gain on both trail runs and bike rides by wearing the watch on a soft fabric wristband.  This improvement would suggest that my issues were caused by sweat/dirt impacting the sensor.

Here is a good example of sweat (most likely culprit) impacting the altimeter.  Both examples are the same run, 2 days apart, similar weather conditions.  It is an out and back run, so there should be symmetry.  Both are purely altimeter based graphs with no correction, showing elevation over distance.

No wristband:

Instinct on wristband (sock):

I've run this route for years.  I know when I'm going up and when I'm going down.  I don't care that there has been a bit of drift over the span of an hour.  The first graph is rubbish, the second is near perfect.

I don't tend to get tracks like yours.  Normally the highs and lows are the same or very close.  The issue I experience is with smoothing which then leads to an under reporting of elevation change before correcting in Garmin Connect (as stated before I did not need to do this using a Fenix).

The difference between the recorded elevation change when mountain biking is huge but I don't get the same issues when recording a walk/hike or even a road ride (roads tend to be smoother than mountain bike trails and cannot walk as fast).

Its a bit of a thing now that at the end of a group ride/event I always don't go as far or as high as everyone else.  I've got used to it now and just ignore the elevation data that the Instinct records.

I don't wear the instinct on my wrist when mountain biking or road cycling but use the Garmin bike mounts.

Here's an example of what I'm getting. It's a very short track, variations of which I ride very often and I've used three different computers here (plus the Instinct), so I've got a very good field of reference on what I should be seeing here. Unfortunately, I don't have any archived files from my older devices. 

I start and finish in the same place and if there was an issue with sweat, I'd expect elevation at start to be different than elevation at finish. The weather during this was perfect and hasn't changed throughout the ride. 
The green section is off-road and following descent is the same route - the descent should be steeper and there's also a short flat section, which is ignored by the instinct. After a quite steep cobbled descent follows the red section, which is road and the following descent is off-road. 

At the summit, I stretch for a few minutes, so if riding position or something of that sort was an issue, I'd expect the elevation to suddenly jump up, but nothing of that sort happens. It seems to me like there's some kind of a correction mechanism in the watch itself that chooses to ignore certain changes in pressure. 

Are you in contact with Garmin engineers? Maybe they could benefit from your tests if you have data to help them debugging the issue. Although this forum is monitored by Garmin employees, I believe that if you have an issue you should contact support and have engineers investigate.

I have not been able to contact the english support probably due to technical reasons (tried three times, didn't bother again). My local support for whatever reason doesn't list the instinct so I had to call them. All they told me was to RMA the watch. 

I really don't want to deal with the RMA process, as I don't think it's very likely they'd give me a refund and I don't want to spend weeks or months without a computer, trying to solve an issue that is probably down to software or hardware design. 

I agree - a new watch will likely have the same issues.  If you are wearing the watch on your wrist, try an absorbent wristband under the watch.  I was seeing the exact smoothing issues you indicate in your post above, and the wristband has resulted in dramatic improvement on trail runs, and road rides. 

Of course, this does not explain the results seen by "tafuk2" who uses a bike mount on trail rides (water, mud, wind exposure perhaps??).  Like "tafuk2", my GPS distance is always under-reported for trail runs in heavy tree cover.

I'll have to try putting the unit on bars or in pocket, soaking in the sink hasn't helped. The elevation deltas were normal / comparable to other devices when the unit was new. Tried a factory reset and that had no effect. Kind of annoying, why bother spending the extra $ for a tracker with a barometer if it isn't reliable.

While I was having much better results by wearing the Instinct over a wristband, my most recent ride was definitely back to major elevation smoothing:

Edge 500: 43.14 miles, 3,907 ft climbing  (I think the Edge picks up a little too much noise, probably 200-300 too high) 

Instinct: 42.2 miles, 2,799 ft climbing (definite smoothing issues on this ride (800-900 too low)

The big differences come with rolling hills, as the Instinct just seems slow to react,  Thus, elevation gain is really lost in the rollers.  Also interesting to see the lag in elevation vs. heart rate (perfect match with the Edge, but HR is already dropping before hitting top of climbs with the Instinct).  

Perhaps a soak will help, as I did a very hard run in hot weather the day before, with tons of sweat and no wristband.

Edge 500: HR is perfectly synchronized with all climbs - big and small.

Instinct: Hard to see here, but my HR is already dropping before the elevation profile tops out on climbs.

My old Edge is pretty amazing at picking up the correct starting elevation (537 ft.) and returning to within about 5-10 ft. of that the end.  The Instinct again shows a little lag and appears to stick at a slightly higher elevation at the end.