It may not be possible to generalise to a "usual" cause, but taking your examples in order:
“shell shock” during World War I, followed by “battle fatigue”, then
“operational exhaustion”, and finally “post-traumatic stress disorder”
I think Carlin's initial description of how euphemisms come about is broadly accurate. People don't like to face the truth too starkly. However, "treatments" for shell shock during WWI included execution for desertion. There was no golden era of excellent treatment of combat veterans, that has been eroded by euphemism.
The switch from "shell" to "battle" appears to be a generalization for the sake of accuracy. Battle in WW2 was less frequently characterised by prolonged shelling, so the WW1 theory that the effective cause was long-term exposure to explosions lost credibility. Observe that Carlin subscribes to this WW1 theory of causation, that excessive sensory input to the nervous system causes it to "snap". This is not correct, and it does not characterise the condition. That is why doctors don't use it.
The change from "shock" to "fatigue" might well be a euphemism.
"Operational exhaustion" appears to me to be a euphemism and I can't immediately find anything to suggest it refers to different diagnostic criteria or a different model of cause. It's a less impressive name for the same thing.
So what about the reasons for these euphemisms? If the term was defined in the military and primarily diagnosed by military doctors, then one would presume (with rather more confidence than is needed for comedy) that the motive was propagandist. It is at any rate in line with a number of other infamous examples of military euphemism: "war" -> "defense", "killing the wrong person" -> "collateral damage", and (arguably) "terrorism" -> "psychological deterrence", "kidnapping" -> "rendition", "torture" -> "enhanced interrogation". Probably it is this kind of euphemism which Carlin is most excised about, since frequently it is both intentional and cynical. The forces behind it, naturally, are political.
"PTSD" is not a euphemism, it's an attempt to generate a clinical term describing the cause accurately (and therefore to include instances of the same symptoms caused by the same mechanism in cases that have nothing to do with combat). It's also taken seriously by doctors, if not always by those funding veterans' treatment. I find Carlin's claim that it hides the humanity slightly insulting, given that I'm one of the audience that it's allegedly hiding it from. It seems to me rather less euphemistic than "operational exhaustion". I could see myself naively believing that "operational exhaustion" refers to the routine need to limit the length of deployments. I can't see myself thinking that PTSD refers to anything other than a disorder relating to stress and trauma. It emphasises that the problem is not inherently the shells, it's the trauma. Now, it may be that people are less sympathetic to mental consequences of combat than they are to physical ones (although again, we can hardly point to exemplary care for physically disabled veterans as the proof of this). In that case it might be that "shell shock", described as a physical injury from shelling, would get more sympathy than PTSD. But Carlin is mistaken to say that's because it's more truthful, it's actually less accurate.
“Idiot”, “imbecile”, “moron” → various mental disabilities
The names of various mental disabilities might have value as euphemisms, especially since the terms on the left are in practice used primarily as non-specific insults rather than as clinical terms. However, that's not what doctors need them for.
"Idiot" and "imbecile" did not originate as clinical terms, they were picked out of the general language and applied as labels for specific IQ ranges. "Moron" I think might be a psychiatric coinage, but lost its specific meaning as attitudes changed to IQ testing and the consequences of particular IQ scores. Doctors need other terms because they're no longer primarily interested in classifying people by their IQ score, rather by the specific nature of their incapabilities.
Now, it's possibly true that a fancy psychological term of the "various mental disabilities" kind, let's say "dyslexia", obscures a supposed plainer truth, and therefore you'd call it a euphemism. You kind of have to pick your side on that one. "Dyslexics? People who are just too dumb to read!" has potential comedic value to the right audience or perhaps even in the hands of someone so elegantly offensive as Carlin. It doesn't have much value in helping dyslexics by researching and teaching specific techniques that help them learn to read. Since doctors are not comedians, they go for the latter. This is because they want to help dyslexics read, and to give them a more specific label than "stupid". It's certainly not because they want to hide the truth in order to dehumanise dyslexics, and it mostly doesn't have that effect. I mostly liked Carlin, so I'd kind of hope that he would not have used this as an example of euphemism the same type as "operational exhaustion", but I'm not so confident in him that I'd bet on it!
“Mongoloid” → “Down syndrome”
This a pretty common pattern, that a medically recognised disorder is named for the person who first successfully characterises it as we currently understand it. Down himself called it "the Mongolian type of idiocy", whence "mongolism". That term is not very accurate, referring to one particular physical symptom with a word in its sense as part of a classification of human races. So replacing the term isn't an act of euphemism. I suppose it could be considered an example of political correctness, since part of the motivation for changing it was its observed offensiveness. The debate over the matter was held somewhat in public, so there are accounts available as to how the term was changed, for example https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1082401/
The WHO apparently dropped the term at the request of the delegate from Mongolia. That would seem to be a legitimate diplomatic choice even in the hypothetical situation that the condition had anything to do with Mongolia or Mongols. Which it doesn't, but nevertheless the "force" behind the change of terminology by that group is observed to be "someone from Mongolia". We can probably rule out "someone from Mongolia" as the typical force behind changes in medical terminology ;-)
“Cot death” → “Sudden infant death syndrome”
The second term seems to have come along within 5-10 years of the first, so probably this is a case of the medical establishment choosing a term in a more formal register. As far as I know this one (unlike "shell shock" -> "PTSD") actually does refer to exactly the same thing. So this might be a case of euphemism, but I suspect actually is just a case of the medical profession wanting to use specifics ("sudden" and "infant"). Removing "cot" certainly avoids the international cot/crib distinction, and I speculate is also considered inappropriate by doctors because it suggests a cause where no cause is known. Since the word "death" is not removed I'm going to say that again this is not a euphemism.
In short, the motivation for euphemism by definition is to conceal an unpleasant truth. The force behind this might be political propaganda or personal discomfort, depending on the example.
However, several of your examples are not of euphemism, they're examples of improved medical understanding allowing for better definitions of disorders, and a desire for names that reflect the better-defined condition. The force behind that is scientific reasoning. You've also got one example, "cot death", which quite likely is avoidance of slang in favour of more formal and somewhat more precise language.