Would it be ok for China to set up military bases in countries near to America, would it be a problem

Why not? The Yanks and Ruskis do it.

Of course it would be a problem!

It doesn’t bother me but yes Cold War era politicians would absolutely take offense and be expected to cause trouble.

It would certainly be very worrying.

The nearest for China would be in a Sino-sympathetic country in South America; but China has other ways to attack the USA (and anyone else) without even leaving Beijing.