I asked a friend why he thought this was, and he suspected that the CS department's focus on formal languages and automatic theorem proving is what lead to their use of logical symbols.
Your friend in math is definitions. IME clever definitions minimize the sheer amount of rigor you need to get from point A to point B through their abstractions. The more "natural" or easily understandable a definition is, the easier it is to use that definition as a ground truth in your theorems.
For example, I have an unpublished proof of a graph/game-theory conjecture. Proving the theorem's correctness is extremely convoluted if you rely on atomic definitions of graphs, valid actions, etc. However, as you define new relationships precisely, it becomes much easier. The more abstractly you approach the problem, the simpler the problem becomes, given the correct abstractions.
Humans can infer. If it doesn't make sense, they realize something is wrong and attempt to make sense of it.
As to why the mathematician preferred 'English', I think it's because quantifiers aren't universal. Mathematical notation itself isn't universal. That is, while the student may be internally consistent and exact in their qualifiers, they are not universally consistent between students, nor with the professor's expectation. The professor wanted to immediately grok what the student was trying to do, not have to approach every single turned in assignment as though it was an unfamiliar mathematical paper, using its own notation, that he had to interpret (where some symbols may share a standard meaning across papers, others won't)