Is asking questions with very short answers on major exams recommended?
There's a bit of a false dichotomy in the way you present things. Exam questions can have a long answer without being hopelessly vague and subjective. In college-level math, one usually can't avoid asking some questions with long answers, whether they're problem-solving, proof-based, or conceptual. Partial credit is just part of the deal, as is handling the occasional student seeking more partial credit. (You have to at least hear them out, because it's always possible you've made a mistake.)

However, there are different views on this; I once had a professor who gave an entirely true/false final exam, with no explanations or partial credit, in an abstract algebra class. His reasoning was that over an exam with say 40 questions, the effects of random guessing would average out to a roughly equal small bump for everyone. I was skeptical, but after taking the test I was confident that the same people did well on it, as would have done well on a traditional "solve this problem, prove this small fact, etc." exam. But this may have worked because abstract algebra has a lot of questions with a concise statement and answer, but that require real understanding. (One T/F question I remember: The Galois group of the fifth cyclotomic field is cyclic.) I don't think it would work for every course; at some point in your math education, someone needs to read the gory details of your solutions and give you feedback.

By all means, look for opportunities to make good questions that can be graded quickly and objectively, but a lot of the time, there just aren't many questions like that.