Very interesting case.
Originally, "s'avérer" means "to turn out to be true" (it is the meaning of latin "ver") and couldn't be used with an adjective: "la prophétie s'est avérée" = "the prophecy turned out to be true". This usage is now considered old-fashioned, if not archaic.
With time, it became accepted to use it with an adjective, like "ce médicament s'avère efficace contre la COVID" (side note: "COVID" is supposed to be feminine, but many people use the masculine).
Despite this usage being already far from what the etymology of the verb should allow, most linguists consider that two uses should still be avoided because of the etymology (but most people don't care):
- "s'avérer vrai", because it is a pleonasm
- "s'avérer faux", because it is non sense
So synonyms of "vrai" and "faux" should also be avoided obviously, but where is the limit? Here it says that "Les prévisions se sont avérées erronées" is a mistake, but does "erronées" really means "fausses", and if yes, what about "incorrectes", and pushing further, "incomplètes"?
There is another verb, that probably influenced using "s'avérer" this way: "se révéler" + adjective:
- "La prophétie s'est révélée vraie"
- "Ce médicament se révèle efficace contre la COVID"
About "il s'avère que" with the impersonnal "il", I suppose that "il s'avère que c'est vrai / faux" is as wrong as in the previous case. And you cannot say "il se révèle que" from what I know.