Re: fully manual
Hmm, I'm trying to think of the mechanism for that being an explanation.
My experience when systems have lost some gas is that they reduce in cooling ability - but don't freeze up (seen it several times). In normal operation, hot compressed gas enters the condenser, is cooled, and at some point it starts condensing (at which point, it doesn't change temperature any more unless part of the condenser contains only liquid, where it can cool some more).
As refrigerant is lost, the quantity of liquid will reduce - and initially little will change WRT performance unless the system is already up against its limit of condensing gas due to the outdoor temperature. There would be a slight reduction in the part of the condenser which only contains liquid, and that would only make a small difference to the temperature of the liquid line back to the indoor unit. As long as there is still some liquid phase, then the system pressures would be barely affected.
At some point, there will no longer be enough refrigerant, and the liquid line will start to contain a mix of liquid and vapour. Here's the clever bit (one of the few bits I recall from doing this at university [cough] decades ago), while the volume flow rate through an orifice is higher for vapour than for liquid, the mass flow rate is lower. Thus a simple orifice or capillary tube can be self regulating - if there's vapour in the line, the mass flow rate reduces, so allowing the compressor to "catch up" (if it has the gas to do so).
So the mass flow rate has reduced, therefore so will the heat input needed to evaporate the liquid also reduce - hence the system will suck less heat out of the air, and the coil should run warmer.
[lightbulb moment]And now I see a mechanism for what you've observed. If the system is an old style on-off constant speed compressor, it may (depends on compressor design) draw down a lower pressure on the vapour side of the system. At a lower pressure, the liquid will boil off at a lower temperature, and part of the evaporator coil will therefore be at a lower temperature.
How that manifests will depend heavily on the design of the coil. In some designs it will allow that part to freeze up with the result that the refrigerant will need to move a little further along to boil off - and so the coil will slowly freeze up from one end. In other designs, while a little bit may freeze up, the already boiled off (and warmed up elsewhere in the coil) vapour will come back along a different loop and control the freezing.
A modern inverter design shouldn't have this problem at all as the compressor should run slower to maintain the same low pressure as the mass flow rate reduces - hence no change in the boiling point of the refrigerant.
Doesn't change the fact that fi the compressor fails to run, there's no way the system is going to freeze up because of that - it's far more likely that the compressor not running is a result of whatever else has caused the system to ice up.
I might add that A/C service engineers are (like service engineers in any other field) not averse to providing "interesting" ideas for why something isn't working. I once had a call from a client at a previous employer where the A/C in their server room wasn't working properly - it had all the symptoms of a lack of gas (turned on, didn't really cool much, then tripped). Bear in mind that it had been working fine for 2-3 years before the service engineer told the customer that "it can't possibly work in that room because it's not insulated and the heat input through the ceiling is over-powering the unit". There was actually insulation in the ceiling (I'd had to go through it when running network cables), and they'd been out to e builder's merchant and bought a roll of glass fibre to add to it. After a few days of this "it's not working, it used to work", "it can't possibly work because ..." to and fro I called the company and spoke to their service manager. I described the setup, the fact that it had been working for a couple of years, and the symptoms - and he instantly agreed that the diagnosis "didn't sound right". An hour later he called back to say they'd found the problem (a stuck reversing valve) and fixed it.