A disposal doesn't make grease disappear. It grinds food into a slurry, and any fat mixed in cools and sticks to the drain wall a few feet past the trap. KC Water runs a Fats, Oils, and Grease program for this reason, since grease from homes clogs private lines and the roughly 2,800 miles of sewer mains serving the city. Ground food catches on that grease layer and builds a plug.
Older homes in Brookside and Waldo add another problem. Many still have galvanized or cast iron kitchen branch drains, and decades of corrosion leave the inside rough. Slurry snags on those rough walls. KC Water reports an average hardness near 100 mg/L (5 to 6 grains per gallon), which is considered moderately hard, so mineral scale adds more texture inside older pipes. If your sink drains slowly even with a working disposal, the trouble often sits downstream, and drain cleaning clears what the disposal can't.