It is the difference between present progressive and simple present.
"So what are you doing nowadays, Don?"
"I am teaching English."
You taught English yesterday, now, and tomorrow.
As to your annoyance:
Your understanding of the progressive tense is that you have to be doing the thing right that very second, but that would mean we would never say "am running" unless we are on the phone panting because we are running while talking.
As you can see, the posed question asked what his profession is and was, what he has been doing, so, at this very moment, and in the past, and in the future, he was, is, and will be a teacher of English.
"What do you do, Don?"
"I teach English."
Simple present--right now, I teach English.
As for "to teach", "to teach" is an infinitive that can act like a noun, adjective, or adverb.
"I plan to teach English." (Direct Object)
"To teach English is my goal." (Subject)