>on the sides
Yeah, you're going to need a trench on the starting side that's at least as long as the width of the sidewalk.
They make a sewer jet for pressure washers that split the water direction. One points forward to clear or bore and the rest are directed to the rear to provide propulsion.
https://www.amazon.com/gp/product/B07RPY26GZI've used these a few times on commercial properties as well as my home. Work very well. The surprising part is they develop enough power to drag the hose along, too. Since you have rental property you might want to add one to your bag o' tricks. Cheap.
For a horizontal bore under a sidewalk, I'd start with the auger, maybe adding an extension to bore as much as possible with the drill. Then I'd drive a 1" or 1-1/4" piece of pipe the length of the it roughly the width of the sidewalk in the bore hole. I'd put the sewer nozzle through that pipe sleeve to finish the bore, tapping the end of the pipe with a mallet as the jetter worked. The idea is that the jetter will flush out the dirt through the pipe.