According to the hypothesis that utility maximization is always a second order effect, it won't affect policy unless it can make the current government lose.

I absolutely think the movement of people and companies can make "the current government lose," often for better, sometimes for worse.

Also, as far as I can tell companies "threatening to move" or "moving" are almost entirely about tax avoidance, and active legislation to handle this (so that profits generated in the country stay taxed in the country) would pretty much get rid of the issue.

No, this issue will never be eliminated, nor would we want it to be. Moving often means profits are no longer generated in the same place - this needn't be just a paper change. Nor taxes the only thing that matters, people and companies also move based on the regulatory regime and provision of public goods.

Other than companies moving labour intensive production to countries with cheaper labour, which theoretically doesn't hurt the country at all.