Objectively speaking; those were some bad tests for leadership. First, strength has nothing to do with being a good leader; some of the best leaders in world history were weakened, like Franklin Roosevelt. Second, the intelligence test was far too easy. The third test was also easy, but it did test one good leadership trait, that of judgement. Even with those traits, one can still see a leader going bad. I'm sure that all sorts of bad historical leaders could have passed those tests. A better test would have been asking how Littlefoot would react to difficult situations, such as a drought or a flood, a food shortage, a predator attack, or a fire. Those qualities are what's needed for a leader; moderation, courage, and wisdom. And a system of checks and balances will still be needed. That, and a respect for all the members in society.
EDIT: The test could have looked like this if it were teaching things practicable to a leader of a herd of dinosaurs:
1. Moderation test: A limited number of leaves are brought out. They have to be shared with others. Let's just assume that Littlefoot's friends are brought out to participate in this test as well, in order to simulate a herd. The simulation would involve a leader having to be moderate, share the limited food with others, and quite possibly, go with less food in order to feed the others. By showing moderation, one won't give in to excess, and thus, will avoid the excesses of tyranny.
2. Courage test: This one is harder to come up with. I suppose that some simulated threat could be given (like the "tar-log" trick we see in the first movie, or a shadow of something that could be frightening, etc.) Anyway, once the simulated threat is brought out, Littlefoot would have to endure it, get past it, and prove to be courageous. This really isn't easy to test without tools or technology...
3. Wisdom test: A series of questions designed to test character. To determine how moral a leader one could be. Some questions could include the question "Is something good because God says it is, or does God say that something is good because it actually is good?" (if the dinosaurs in this series have a conception of God, that is.) This would test whether the potential leader believes in the false maxim "might makes right", which a good leader will NEVER believe in. Another question could be "what is the wisest thing one can say?" This can be answered with "The wisest thing one can say is how little they really know." By admitting how little one
knows, one won't make the mistake of assuming that one knows what one does not know, and will therefore be able to see reality all the more clearly.
Of course, true political virtue cannot be taught; it can only be learned. This is why there is no such thing as a "president's school." The wisest thing to say about political virtue tests (what the tests in the show were), is that they are all insufficient.