Over time, does the remaining piece of root come to the surface by itself?

Sometimes it does, sometimes it doesn’t. If the
root fragment is firmly lodged into the bone, it will remain there. If it is loose, it may work its way to the
surface. Either way, do not stress over
it. It is not uncommon for a “small”
portion of the root to fracture, but it is rarely a problem.

It is true that your remaining tooth piece will either work itself out. Or, it will heal over with bone. There is no harm leaving a small fractured piece of tooth. In fact, trying to get that piece out can push it up into the sinus, which is then a much bigger problem!