Learn English – Is “Whom should I give this job to?” grammatically correct

The Wikipedia link about the accusative case explains that

Modern English, which almost entirely lacks declension in its nouns, does not have an explicitly marked accusative case even in the pronouns. Such forms as whom, them, and her derive rather from the old Germanic dative forms, of which the -m and -r endings are characteristic.

Now, whether to use who or whom in your sentence entirely depends on which case should be used, accusative (whom) or nominative (who).

In English, it is grammatically correct to use nominative after the verb to be as in

It's he who stole my car. It's they who told me the truth. It's she who lied to me.

However, we know that "It's me" (using the accusative case after to be) is broadly used in English. But it is just a few exceptions.

In your sentence, it is appropriate to use the nominative case as it is the complement of to be. If you divide the sentence into two parts:

He must decide / He should be who => He must ask who he should be => He must decide who to be.

in the same way as:

He must decide / He should meet whom => He must decide whom he should meet => He must decide whom to meet.

He must ask / She is who => He must ask who she is. (This question cannot be shortened with wh-word + to-infinitive as the subjects are not same.)

We don't ask,

*Whom is he? or *Who is him?

*Whom am I? or *Who am me?

because whom and him/me are the accusative case and can't be a complement of the verb be in this case.

Note: "He must decide who he wants to be" is more idiomatic than "he must decide who to be".

I don't think that there is a relevant difference between sentences like "He is demanding £5,000 from the elderly woman whom he claims has ruined his life" and "Whom should I say is calling?" If there is a relevant difference, it would probably be that "whom" is a relative pronoun in the first, and an interrogative pronoun in the second. Possibly, interrogative whom is less common than relative whom, but I haven't seen much literature that discusses the evidence for that idea in detail.

The word whom is fronted in both sentences. The 2012 Language Log post "Sometimes there's no unitary rule," by Geoff Pullum, refers to "a preposed relative or interrogative who" when discussing this topic.

However, JK2's answer points out that in Chapter 5 of the CaGEL (published in 2002) the use of whom in this context is described as being mostly restricted to relative clauses. I don't know whether Pullum's views have evolved since the publishing of the CaGEL, or whether the inclusion of interrogatives in the rules given in the 2012 blog post was an accidental oversight.

There is a 2004 blog post by Pullum ("I really don't care whom") that describes the general state of the who vs. whom distinction as confusing and says that whom "hardly occurs in interrogatives at all".

Whom would not be an "object" here

It's not correct to categorize whom as "the object of the whole clause", any more than he is the object in sentences like I know that he is calling. Rather, it is "not the subject" of the relative clause. Hopefully the distinction I'm making between "not the subject" and "object" is understandable, if nitpicking.

Whom is "incorrect" here from a prescriptivist perspective

The sources that you have seen are all taking the prescriptivist viewpoint. The prescriptivist rule isn't inherently invalid; the point that descriptivist linguists like Pullum are trying to make in documents like the linked blog post is that the pattern of usage that the prescriptivists condemn has a logic of its own, and it seems likely that many people who write things like "the person whom I say won the race" aren't just accidentally using "whom" where they say "who" (i.e. it's not just a typo or "production error"), and aren't just consciously choosing to write "whom" because they think it is more formal and they don't know where it should be used (that kind of thing is what linguists like Pullum would call a "hypercorrection"), but rather are following a rule that they have internalized to some degree that says to use "whom" in contexts like this. So it is "grammatical" for them in the sense that it is consistent with a rule that they plausibly have acquired as part of their own personal system of grammar.

From a descriptive viewpoint, things that are consistent with a speaker's grammar are not described as "mistakes" or "errors". "Acceptability" depends on the grammar of the listener/reader. The articles that you have seen use words like "mistake", "wrong", "incorrect", "error" in a different sense, to refer to things that aren't consistent with the rule that prescriptivists have historically preferred. For example, from a prescriptive viewpoint things like "ain't" or "haven't got no" are "incorrect". This is a common use of these words in popular discussions of grammar, but modern linguists tend to avoid using these words this way in scientific contexts.