I had two brothers. The one who is now in a better life used to say “No problem” all the time.
Make no mistake, he was fully aware of the non-subtleties you have pointed out about the phrase. He used it to emphasize the fact that whatever he was doing for the other person ~was~ a problem. Usually a damn big problem.
But for some reason, that brachycephalic cretin seemed to think he was entitled to some service that was a gross imposition at best and utterly rude at worst.
“No problem” actually meant (to my brother) “You’re a colossal pain in the ass, but I’m going to be the better man about this and pretend to make a civil response.”
Now ..... I’m not saying that the folks you’re running into are quite that convoluted in their reasoning, nor that you would ever so much as border on being “a problem” to someone serving you.
I use the phrase myself sometimes .......
“I use the phrase myself sometimes .......”
No problem.