I am inclined to agree with @josh that many of the examples @user178049 gives are not really plural-only nouns: apology, compliment, condolence, in their plural forms, really do just refer to multiple instances of their singular forms. "Thanks" is a weird one, and also, uniquely amongst the examples we've seen of what @user178049 elegantly summarised as "words of compensation and reward... and formal expressions of feelings", it's not an abstract noun derived from a verb by adding a suffix like -tion, -ence, -ing and then pluralising it.
Lots of interesting points have been raised, but to my mind the answer must be connected to how "congratulations!", "commiserations!", "thanks!", "apologies!" etc are reduced from versions where there's a preceding "many" or "my". Just naming a singular noun as a one-word sentence would feel very weird indeed -- do we ever do this? My gut feeling is that single-word sentences using a plural noun are only valid because we have generalised the rule that you can omit the "many"/"my" etc from the two-word version, but we haven't extended that to singular nouns.
It still leaves me wondering why we use the plural forms in the first place: "please send her my apology", "he sent condolence" etc are grammatical, but they give the impression that a single, specific message is being sent. But when we hear "my apologies", "his condolences", we conventionally understand it to mean the metaphorical extension that it's referring to a sentiment, not a specific message.
Thesaurus.com says that "congratulations" has been used in this manner since C17 but doesn't cite sources