Is “completely good” proper English? If not, why?

Perfectly good would be different than saying completely good in most contexts: perfectly good tends to be used in a hyperbolic sense, meant to signify something that's "good enough". For example, that apple is perfectly good to eat is not the same as saying that apple is completely good to eat: the apple might have some flaws, but none that would prevent it from being eaten.

Which leads to the point that you'd only want to say something is completely good when there is no chance for it to be bad. Completely good is not a good substitute for very good or really good, which merely mean that something is notably good, not entirely or absolutely good. But, the Christian God or a saint would be characterized as being completely good because they aren't even remotely evil by definition.

Finally, if you wanted to say something was good in the superlative sense, completely good would be wrong. The correct word would be best, as in that course of action is best not that course of action is completely good.

"Completely good" is grammatically correct. As has been said, "completely" is simply an adverb modifying "good." The phrase only sounds odd because it is not commonly used.

It seems to me that as long as you are using good and well in their proper locations, you should be fine.

For example, good is a noun or adjective, and well is an adverb or adjective. You can say "He is good" or "He is well," but when you use a verb like "to run" then you have to say "He ran a good race" or "He ran well." See here, well is an adverb, but good is not. "He ran good" is improper English.

As long as this is observed, "completely good" should be fine.