Is this really a real thing? I mean, has anyone, any group or even the Chinese government ever expressed itself negatively about it? Or is it just the case for writers or filmmakers being overzealous about it?
Yes, no doubt in the beginning he was an example of the "Yellow Peril" cliché. But as has been mentioned here on this board recently, there have already been attempts to get him away from it.
We had the crime boss version Gene Khan, wearing a suit. We had as well, the Dave & Bob’s version, that is quite similar and still has the bonus of giving him a suit of armor. And we had yet another suit-wearing version that appeared during the Knaufs run.
None of these versions lasted long, but if they had lasted and were developed, would the problem have ceased to exist?
It's been a very consistent thing with China's government that they do not allow Chinese villains in foreign movies. I don't know if they commented on The Mandarin specifically. I would tend to think not, simply because Marvel nipped that possibility in the bud before the movie was even made.
(A minor nitpick: they probably would not have a problem with his ethnicity, only his nationality, since he's not Han)