Can you ever use the continuous aspect with “usually”?

Solution 1:

Your teacher is right. However if you add a time expression at the end of it what we indicate is different. Compare;

I usually go to school at 9:00.

I am usually going to school at 9:00.

The first one tells us the action starts at 9:00.

The second one tells us the action started before 9:00 and you will be in the process of that action.

A: Can I call you at 7:00?

B: Umm I am usually (in the middle of) eating at that time so how about 8:00?

There is also "always" usage which means: doing something very often, too often, more than normal, which has a negative meaning. Never heard it being used without some kind of complaining actually.

I always drive to work. I do it every time.

I am always driving to work. I should ride a bike and save some money.

A few more examples;

You are always watching TV. Stop it and do something else!

My wife is always complaining. She never stops!

You are always playing video games! Go outside once in a while!

Solution 2:

There is an explanation in Murphy's "English grammar in use" Unit 3:

We use simple for things in general or things that happen repeatedly:

What do you usually do at weekends?

So your teacher is correct.