Suppose the moon rotated on its axis just as quickly as Earth. Would we still always see the same side of the moon from Earth?

No, The Moon, on the other hand, rotates once around its every 28 days, and once around the Earth in that same 28 days. The result of this combination is that the same side of the Moon is always facing the Earth.

There is an important difference because the solar energy is provided by the sun and matter doesn't decrease from Earth no matter what. The solar energy input provides plants with energy to grow and for animals to not die of the cold. Plants are food for omnivores and herbivores and thus supporting the earth's ecosystem.