Ergonomic
A tuner using an L-lever must apply a strong but precise force to the tuning pin from a cantilevered position of severe mechanical disadvantage. When tuning a grand, the tuner must must reach up and over the stretcher, and use the muscles of the hand, wrist, arm, and shoulder to apply force to the pin; and when tuning a vertical, must work with the tuning arm hanging unsupported in front of the tuning pin field.
By contrast, a tuner using a C-lever to tune a grand can work with the arms and hands in a relaxed, natural position above the keys, using the strongest muscle group in the body--those of the torso; and, when tuning a vertical, can stabilize the tuning arm on the top surface of the piano's back.
Convenient
To tune the highest treble strings on a grand, a right-handed tuner using an L-lever must either begin to work from the treble side of the piano, or reposition the lever to nine o'clock and switch hands.
By contrast, a tuner using a C-lever, whether right- or left-handed, can tune the highest treble strings of a grand in exactly the same way as all the other strings.
Because L-levers always flagpole tuning pins, but C-levers don't, the fit of the tip to the pin and the tightness of the pin in the block have a significant impact on the ease with which a piano can be tuned with an L-lever, but very little impact on the ease with which a piano can be tuned with a C-lever.
A tuner who using an L-lever who sits to tune grands needs a tall bench. By contrast, the height of the bench makes no difference to a tuner using a C-lever.
Finally, a tuner who uses a C-lever to tune a grand doesn't need to clear the lid in order to work.