Hope couldn't maintain a screen career because he could never do what any successful screen actor HAS to be able to do: let the audience in, show at least a hint of vulnerability, of warmth, of empathy. His presence as a comedian was defined by impersonality, by distance, by refusal to connect with anyone.
It's why, whenever he introduced a beauty queen or Heisman trophy winner or decorated soldier or Republican politician or random ordinary human on his tv specials. he always sounded like he was reading the name- even if it was part of a scripted sketch- off a piece of paper someone had handed him ten seconds earlier, and you had the feeling that, if he ran into the person he'd just introduced backstage two minutes later, he'd have no idea who they were. It was the comedy of coldness.