A road-side patch can be done without removing the wheel.
I have gotten pretty good at levering out one side of the tire, pulling out 1/4 of the tube and checking for a leak, stuffing it back in, and rotating the tire, and checking the next quarter of the tube, until the leak is found. I patch it and then stuff it back in. Then I lever the tire back on the rim, inflate halfway, let the air out, inflate all the way, and drive.
I did get tired of even doing this and put flat-out sealant in both my tubes. The thin factory tires were replaced with thicker Innova Hybrid tires. No flats since then.