What matt_mcl describes is the key difference between descriptive and prescriptive grammar in spoken language. Although revealing somewhat of a bias against prescriptive grammar, it is essentially correct.
But the OP described a situation in which students who were good writers were struggling with a teaching-English class. The problem here, as referred to by hazel, is that of written language, which has rules that are often different from, or in the case of mechanics, irrelevant to spoken language. Spoken language doesn’t have punctuation or spelling, the rules of which must be learned. People who read a lot have been exposed to a lot more properly punctuated written language than those who don’t read much, and so have internalized many of the rules, which they can then apply to their own writing.
But English teachers don’t necessarily have to be great writers. Because they are going to be teaching writing, they need to be able to identify the good and the bad in their students’ writing, and be able to articulate to their students exactlty what the problem is and how to fix it. For this, the teachers do need to have a solid fundamental understanding of the prescriptive rules of written language: diction, syntax, usage, mechanics, and spelling.
A useful analogy. A person can drive a car, and drive it quite well, while understanding very little of how the car works. To operate the car, one only needs to be able to use the ignition switch, steering wheel, accelerator, brake, clutch, and gearshift. One doesn’t need to know why or how these things work, only how to operate them. Likewise, knowledge of how any of the numerous, complicated systems in the car work is not needed to drive it.
But if you want to fix the car when something goes wrong with it, you need to have much more in-depth knowledge of how the car works.
Likewise, a person can write, and write well sometimes, without understanding the intricate workings of the language, but one cannot be a good editor without such knowledge. And since English teachers are by definition at least partly editors (they need to evaluate others’ writing, be able to identify good and bad parts, and give coherent advice on how to fix the problems) learning the formal rules well should be required.