Some tires are easier to get back on than others. You may not be able to reinstall a difficult tire without removing the wheel. Finding the cause of the puncture is sometimes easy, sometimes difficult. Finding the difficult culprits is much easier with the wheel off the bike. With a large puncture or tear at the valve stem, a patch may not work. In the wet, a patch may not adhere. You may need to replace the tube, which requires removing the wheel.

I always remove the wheel for reparing a flat, but don't use a belt drive or IGH.