Every non-native English learner tries to imitate natives, to the best of their ability. I don't think there's any other way to learn. Some people relax and stop trying to improve once they notice that they get to a level where they are understood, though. And then, one can be aware of one's limits (I don't think I'll ever get to distinguish "ice" and "eyes", so I don't even try anymore).
What you describe seems to be some kind of linguistic "uncanny valley" effect.
I was chatting to a French-born colleague and he mentioned visiting the camping store "Millets". He pronounced it as a French word (something like "mi-ey") and I had no idea what he was talking about. It was unusual as otherwise his english was flawless.