Yes, the Roman Republic should have been reinstated after the assassination of Caligula, thus preventing Nero's reign, because there were so many good things about the Roman Republic. Any way that the Roman Republic could have kept going should have been explored. It was strong economically and it was managed well.
If history has taught humanity anything consistently, it has been that power should never be concentrated with just one person. Across cultures, continents, races and even centuries, history is replete with examples of how civilizations decline when in the grasp of one over-powerful leader, but possibly thrive when there is a broader ruling class who has a self-interest in maintaining order and prosperity
I definitely think that the Roman Republic should have been reinstated after the assassination of Caligula, thus preventing Nero's reign. But of course, hindsight is 20/20. I think that at the time, the people still believed in having an emperor rule. They just had no idea that Nero would have been such a destructive force.
It is impossible to say rather or not an alternative event in history may be been beneficial on the whole. Therefore, it does not good to assume that the Roman Republic should have been reinstated after Caligula passed. We assume that this would prevent Nero's reign, but there's no guarantee that is what would happen.
The Roman Republic had been created by great men, and it would have only taken moderately competent men to maintain it. However, the system of leadership appointment used by the Romans proved to be absolutely corruptible. Caligula and all his eccentricities is the perfect example as to why the Roman Republic should not survive.