Assuming "there is a shortcut to the city". Does that mean I could write: "This shortcut is to the city"?

Those two sentences have different meanings. The former means "there exists a shortcut to the city." The latter means "this particular shortcut (to which I am referring) leads to the city."

For some reason, I feel that for a noun phrase with a preposition modifier following the main noun, the corresponding sentence constructed by inserting "be" between the main noun and prepositional modifier is slightly awkward: "the answer to the question" --> "the answer is to the question". "the assistant to the manager" --> "the assistant is to the manager".

It's not just awkward looking--it's incorrect. When you add the "is," you're turning "a noun phrase with a preposition modifier" into a complete sentence that cannot have the same function.

It seems that the "an obligation to his family" --> "His obligation is to his family" transformation is okay, but "an assistant to the manager" --> "The assistant is to the manager" transformation is not okay. Why is that?

Neither of those transformations work to provide direct substitutions. "His obligation is to his family" is a complete, self-sufficient sentence all by itself. "His obligation to his family" cannot function all by itself--it must be either preceded or followed by something. To illustrate this: "His obligation to his family was a heavy burden" or "He was burdened by his obligation to his family." You cannot replace "his obligation to his family" with "his obligation is to his family." See:"He was burdened by his obligation is to his family." It doesn't make any sense.

Let's forget about the transformation thing. I screwed up on that. But anyway, why is "His obligation is to his family" okay, but not "This assistant is to the manager" or "this answer is to second question" or "this key is to the front door"?

It's ok--I think I was misinterpreting what you were trying to say. Anyway, "The assistant is to the manager" or "the answer is to the question" are both valid sentences but I agree with you that they are slightly awkward. This is due to fact that they can be expressed more succinctly: "the manager's assistant" is a far more common way of expressing this. As for "the answer is to the question," "question" is assumed since there is no answer without a question (unless the answer is 42), so we are left with "the answer."

Yet, somehow, "his obligation is to his family" is excellent English?

Well, excellent is a strong word, but yes, this would be a well-accepted sentence. The expression is "to be obligated to [someone, something, or some action]" where to is part of the expression. The sentence "his obligation is to his family" is somewhat of a rearrangement of this expression.

Would that reasoning work for "His duty is to his family"?

It absolutely would.

But "duty" does not have any corresponding adjective form.

Dutiful is the adjective form of duty and you can be dutiful and "dutiful to something."

But "assistant" could be used as an adjective. Yet, you object to "This assistant *is to the manager".

Look, there aren't always hard and fast rules when it comes to language usage. One's decision to use a word or phrase is based on what's generally accepted as proper grammar AND/OR what is common amongst a group of that language's speakers. My position on the "is to" isn't black or white. I believe that the instances of "is to" that I said are valid would be considered to be within the constraints of english grammar. However, there are some usages of "is to" that would sound awkward to my ear because it deviates from the norm within my language group.