Why sing with Solfège?

It teaches function. Sol to do is a very important relationship in any key, major or minor. Without solfege or numbers, I cannot be sure you understand what you see. It is a window to your brain.
It makes sight reading easier for those who know the system well. Perhaps we will never learn the system well, just as adults we will never learn a foreign language without an accent, but we should still teach children a foreign language.
It is a very efficient method to teach music, e.g. to correct mistakes at a rehearsal or lesson. (We are all teachers even if we do not work at a school.)
It is a practical skill. It may land you a job or a T.A. (teaching assistantship).

Numbers accomplish many of the same goals, but solfège is more musical. "Seven" is two syllables and several numbers end with consonants. Solfège also works better when accidentals appear. On the other hand, everyone knows numbers.

Why sing?
From Leonard Bernstein — In the olden days, everybody sang. You were expected to sing as well as talk. It was a mark of the cultured man to sing.
From a jazz musician — If you can't sing it, you don't know it.
From Teaching Approaches in Music Theory, 2nd ed. by Michael Rogers, pp. 127 and 128:
	To be able to sing—to develop a well-supported, controlled voice that is reasonably attractive and accurate—is one of most useful tools of practical musicianship. For any teacher or conductor it is almost indispensable for quick demonstrations of style and interpretation, or of pitch and rhythm. For just simple communication with another person it is often handy or even necessary to illustrate a point musically through vocal means. But for the theory class it remains secondary to internal hearing. For testing—to make oral the aural—singing is the teacher's window into the mind and ear.

P.S. Please learn and teach piano lessons as well.