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The boys, including Gene, actually have the presence of mind to call the wrestling coach for first aid. Moving Finny at this point could make matters worse. I don't think there was any underlying emotional baggage that held Gene back when Finny crashed on the stairs.
He doesn't feel worthy to help him. He feels he has not been a good friend to Finny, and he feels guilty for all the pain he has brought him. He allows another boy to wrap a blanket around Finny, and then tells us "I would have liked very much to have done that myself; it would have meant a lot to me. But Phineas might begin to curse me with every word he knew, he might lose his head completely, he would certainly be worse off for it. So I kept out of the way."