Should car still run on alternator with ground terminal off battery?

What I can gather from your post is this. Nothing. Don't know what kind of car you are working on. Don't know if it is a street car or race car. Don't know if any modifications have been done to the wiring. See the problem?

It sounds like you may have two different problems. Search this forum for "alternator test" or something similar. Member 67Dart273 has posted at least 4,752 times about how to test this stuff. Solve one problem at a time.

And stop pulling the battery cable off the "test" the alternator.