I was recently told by two different people that if tree branches are growing down toward the ground they should be cut off as they are of no use, and are preventing the tree's growth up, and inhibiting the natural shape of the tree.

Is this fact or fiction?

Fiction.

The tree is growing branches where it is logical to grow branches, and the rationale that they're useless or preventing "growth up" is purely human justification for trimming the tree. The reason you might want to remove the branches have to do with your convenience, such as being able to mow under it or open up a path.

If you have a bunch of suckers on a tree that are all growing up from around the roots at the ground, this could be a sign that the tree is planted too deep in the ground. How old is the tree, and what does the base look like? If you can't see a root swell at ground level the suckers might be a sign of stress at being too deep in the ground. In that instance, you would want to use an air spade to clear the area, or in lieu of that, carefully scrape away the soil, even if this means leaving the tree standing in a shallow dish area. It will breathe easier.