Toilet water and laundry water definitely goes to the same place. We are not nearly advanced enough to have separate sewer systems here - all ends up in the same wastewater treatment plant!
BF poo is only 'worse' than formula poo in that it's more liquid, so it doesn't roll off, but it's pretty easy to rinse off (I just did it in the laundry tub, although I do have a spray hose on my laundry tap). It smells nicer than formula poos

(Once you get solids in there you might want to reevaluate how to handle the poos though!)
There's a school of thought that says esp with BF poos, you don't necessarily even need to rinse them off first, you can just throw them in the washing machine and do an extra rinse cycle before washing. You certainly don't need to worry about getting every last bit off in a rinse.
Personally I found flushable liners a bit of a waste of time when JJ was still exclusively breastfed because of how liquid the poos were