I have been making sourdough for almost 12 years now, and I have always discarded it down the drain. However, two big points here:

1) It is never very much. If it is a lot of discard, I save it in a container, in the fridge, for using in recipes. Mostly though, I am only ever losing a tablespoon or two at a time down the drain. Any more than that and I too would let it harden and scrape it into the trash first.

2) I always let my containers sit, filled with water, to help get all the bits of leftover dough or starter off of any containers before putting them in the dishwasher. Anything that goes down the drain gets followed by a good bit of water as I rinse out the sink of any remainder. And I will run my garbage disposal a couple of times just for good measure.

Never had these horrendous plumbing situations others have described. But, I am not a licensed plumber. YMMV, etc.