Have any developments come out of adding the digits? My impulse is to dismiss it out of hand because it only works in base-10, which in my mind leans it towards numerology instead of math.
It looks like there are some applications. Checksum algorithms are probably the easiest to appreciate.
I think small random uses add up to practical value although there are those who make a religion out of its 'meaning'.
some (n mod 9) can be found by (is congruent to) (sum of the digits mod 9) instead is the most obvious example
True only in base 10 although similar congruences exist for other bases and also involve adding the digits
"Interestingness" is often as good a heuristic as any when looking for paths that lead to useful developments, although the path is often not a straight one or short one.
I also like the idea of secondary and tertiary effects. One simple example: By "playing" with cute yet fun ideas that are highly likely to not lead to anything immediately interesting, we can build up skillsets and capabilities that lead to very useful results for other problems. Perhaps this is somewhat akin to how the young of predator species "play" around in a way that prepares them to actually hunt when they are older.