Learn English – Why is ‘He is a man whom I look up to.’ not quite right even though it is not wrong

Why is the first sentence grammatically incorrect? I see two reasons. First, the preposition is stranded. That "with" is logically attached to the "who" but it doesn't appear anywhere near it. Second, if the "who" is the object of "with" then it should have the objective form "whom".

Why is the second sentence an improvement? Since the relative pronoun is omitted, it's not possible for it to have the wrong form. There's no difference between "who" and "whom" when neither one of those forms is actually present. Also, it's hard to say that the preposition's object is in the wrong place relative to the preposition when the preposition's object doesn't appear anywhere.

Why is the third sentence awkward? Well, there's no real answer to that. The grammar found in the third and fourth sentences is the same. The structure is common in the formal registers of many educated dialects, but absent in the informal registers of most conversational dialects.

Sentences three and four are both technically correct, and for exactly the same reasons. The difference is that sentence three is in a context that makes it sound inappropriately old-fashioned and condescending, while sentence four is in a context that makes it sound appropriately old-fashioned and formal.

Your proposal uses the objective form but leaves the preposition stranded. Some dialects find that acceptable, others don't. If you're going to follow these guidelines, then you'll use the pattern of sentence two in ordinary conversation and save the pattern of sentence four for special occasions.

Yes, I realize that sentence three uses a restrictive relative clause while sentence four uses a non-restrictive. That doesn't happen to be relevant to the question at hand, since the relative pronoun "whom" can be used in either case. What is relevant to the question is how the relative pronoun relates to the preposition of which it is the object.

Or, informally, the preposition it's the object of.

In formal English it is correct to employ the objective form whom when who represents the object of a verb.

In the technical language of grammar the term object designates either a) the complement of a preposition (a sense which is not in play here) or b) the complement of a verb which in some sense receives or suffers the action of the verb.

The verb BE, however, does not ascribe an action to the subject, but an identity or quality. There is no object with BE but a different sort of complement called a predicate complement. Because that complement is not an object it is not represented by whom but by who.