Curiosity about pronouns and possessive adjectives with everyone/everybody and everything

As everyone/everybody takes a singular verb, the pronoun should be he/him, she/her with possessive adjectives his and her. But this is only found in formal English. In ordinary conversation the plural forms they/them and their are used instead:

Has everyone got their books?

Everyone enjoys it, don't they?

Everyone likes their own way/ways of doing things.

Everything, however, has the pronoun it, and possessive adjective its.

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My comments:

I confess, even if formal contexts, I have never seen something like

Everyone enjoys it, doesn't he?

Does that really exist? Which option to choose, let's say, for a language proficiency examination?

I shall find it out and bring it here, hopefully...