[subject] + 很 + [adjective] [subject] + 不 + [adjective] [subject] + [adjective] + 吗? [subject] + [adjective] + 不 + [adjective]?
An adjective can be the whole predicate of a sentence. It takes no verb in front of it, and the slot where English puts "is" is filled by 很.
是 links the subject to a noun, and 忙 is not a noun, so 是 has nothing to do here. English "I am busy" and "I am a student" look alike, but here they are two different sentence types:
Drop 很 and the sentence stops being a plain description. 今天冷 implies "today is cold – as opposed to yesterday", and a listener waits for the other half.
不 replaces 很: 我不忙. 不很忙 does exist, but it means "not very busy", and 不太忙 is the usual way to say that.
Questions take 吗, or repeat the adjective around 不.
很 is only the default filler. Swap it for 非常, 真 or 太…了 and the degree becomes real.