Detail: I have a cheap little digital LCD aquarium thermometer with its probe directly touching the indoor unit's temp probe. Its absolute accuracy might be in question, but when you see the symmetry of the problem you might realise it doesn't matter:
Mode HEAT target 16degC: turns on at 19degC and turns off at 24degC (just tested now).
Mode COOL target 28degC: turns on at 26degC and turns off at 21degC (approx from memory, been a while since summer).
These trippoints are really far apart and not at all what I'd expect. I have to intentionally choose uncomfortable looking temperature targets on the remote to get semi-reasonable behaviour. This makes me think I'm completely misunderstanding something.
One thing I would do is get a thermometer intended for air/gas.
An aquarium thermometer is designed to measure the temperature of a large mass of water with a slowly changing temperature, and the water’s high thermal mass and slowly changing temperature means that there is plenty of time for the probe to acclimate, which the water can do quite effectively.
In contrast, gases need temperature sensors with very low mass, because gases are terrible conductors of heat. A sensor intended for liquids can respond very slowly in air, depending on its design.
And if your aquarium thermometer’s possibly large mass is touching the small mass of the AC unit’s sensor, it might be completely slowing the AC’s sensor response by coupling a much larger mass to it. This could totally throw off whatever PID control loop the AC is using.
Now, I don’t actually think the aquarium thermometer sensor is likely to be the source of your problem, but it’s just something I would rule out first.