Originally Posted by Shimagnolo

The location of a puncture is almost always easily identified before doing anything.
Occasionally, it might be necessary to add some air to detect it.
Unless the damage is so extensive that the tube cannot be patched, there is no reason to remove the wheel to patch.
Just pull one bead off the rim enough to pull out tube at puncture, apply patch, push tube back in, reseat bead, and inflate.
It is a hell of a lot quicker than needlessly removing the wheel.