I think the main reason is defense against cancer. Every division is a chance for mutations, and If you had the already-existing cells dividing, then they would build up mutations fairly quickly. However, you can instead have a stem cell divide once (producing one daughter cell and one cell which remains a stem cell), and then the daughter cell divides a bunch of times to produce perhaps several hundred cells.

Any mutations that occur in all of these divisions will eventually disappear because the cells will die and be replaced with new cells – but the new cells will be replaced from the stem cell, which only has to divide once more. Meanwhile, mutations that occur in the stem cell (which is where it really matters) only build up very slowly – because even though it is producing many cells, they all come from the daughter cell, and the stem cell itself only needs to divide very few times.