Speaking of the towel wrapped hose idea......
Hardware stores sell a bladder like thying it is black rubber,, Kind of barrel shaped,, Comes in assorted sizes from 1 to 4 inches (I think that's the range the 4" one is likely 3" but expands to 4.5 or so)
Hook it to a hose. Shove it down the drain and it will expand to seal, then a release valve on the end lets water pass putting full city water pressure behind the push.. I've blown quite a few clogs using those over the years.