For what reasons is the vice presidency so lightly regarded?

The Vice President's entire role is:

1) become President if the President dies,

2) preside over the Senate,

3) cast a tie-braking vote in the Senate.

4) help out with whatever the President lets him do.

Presidents don't die in office very often, so role #1 is not very important. In practice, the Senate is actually run by the Majority Leader instead of the Vice President, so role #2 is a mere formality and not at all important. Most important Senate work is crafting legislation and voting in committees instead of voting on the Senate floor, and even then, the Vice President cannot vote to end a filibuster, which is realistically the only way to get something through the Senate anyway. Most of the Vice President's power comes from role #4. In other Words, the Vice President's power is almost entirely dependent on the whim of the President.