Is the phrase “no problem” a problem ?

You don't have to understand or agree with management decisions. But you do have to follow management policy to keep the job. If you're in the habit of saying "no problem" or "not a problem" then you need to break that habit pronto. Stop saying it. Even when you're not at work, don't say it. 

Instead, say something else. "Very good" would work. When you'd say "no problem," just say "very good" instead. Silly or not, it gives good results: your bosses are happy and you keep your job.

As a regular customer at a cafe where one of the waitresses almost always does say, "No problem", I have to say I really dislike it.

I say, "A flat white please" and she walks away to get it, saying 'No problem". Well OF COURSE it's not an effing problem: I'm in a cafe, aren't I? How can a coffee in a cafe be a problem EVER?

I don't want to hear the word "problem", even if it's NO problem, in connection with my order.

Then when I pay and leave, saying "Thank you" at the door, she says "No problem" again. WHY??????

Why not something simple and pleasant like "Thanks, see you soon!"?

People take a word or phrase the "wrong" way because they didn't grow up saying it. And that's true not only of older people, but also people from other countries, other parts of the country, or even the next block over. When you deal with the public, you have to be aware of such things.

It's a problem for THEM and it behooves you to change that habit.

Your know why your managers are against it. You yourself said they feel it implies the customer is causing issues.

Try "you're welcome" or "happy to help".

I hate that expression as well.  If your job is to serve food (or anything else) to customers, it's obviously NOT going to be a problem to sell them what they order.  Why would it be?

No problem IS a problem. I used to own my own company. If any of my employees used it, I would tell them not to use it. You should say “You’re welcome” or “My pleasure”. Saying “No problem” leaves me feeling like I’ve caused a problem because that’s what it implies. "No problem" is also too slangy and is mostly used by millennials. You don’t want to sound like a kid. You want to sound like a seasoned service professional.

Fast food staff and management aren’t exactly known for their critical thinking skills. There’s nothing wrong with saying no problem unless you are mentally stunted and fixate on the deeper meaning of words in the most infantile way.