“Predicate nominative” pronouns are not archaic as I would define the word. (I’ll use my definition of “archaic: since the question does not provide one.) I would say an “archaic” usage is one that is not generally encountered in ordinary speech or writing at any level of formality, like “thou," “thee" or "panteth."
But predicate nominatives can be encountered in ordinary modern writing and speech. Some may think of them as pretentious, or ungrammatical (from a linguistic point of view), but the fact remains that they exist in current usage, even if they sound bad to some people.
Obviously, this does not mean that they are usually used, or that they can be used in every situation. One notable feature of language is variety. There is rarely only one way to say things, and linguistic structures often can only be used, or are in practice only used, in certain contexts. Predicate nominatives are certainly not used much outside of formal contexts.
One particular circumstance where predicate nominatives are apparently still preferred over predicate accusative pronouns* is in sentences like “It is I who am at fault?” where the predicate personal pronoun is followed immediately by a subject relative pronoun that refers back to the personal pronoun.
*See Barrie England’s answer to the linked question, which references research mentioned in the ‘Longman Grammar of Spoken and Written English’.
The part about "whom" really is a separate question, but "whom" is also not archaic yet. It is still a somewhat commonly selected option in formal contexts in pied-piped prepositional phrases introducing relative clauses, such as "The person of whom I speak" and it may even be encountered in non-standard formal-sounding constructions with extra prepositions such as "of whom I've heard so much about."
There's a big difference in meaning between a statement like "the predicate nominative/whom is not archaic" (which is true) and a statement like "the predicate nominative/whom is always used in all circumstances where it was traditionally prescribed" (which is false, and has been false for longer than you would probably expect).