All traditional pitchers perform this injurious mechanic recovery position because of the preceding centrifugal mechanic. Injuries you can expect are, posterior labrum tears, anterior labrum tears, bicep and bracialis tendon pinch and overstress where it attaches through the edge of the labrum and into the glenoid fossa, deceleration muscle tears in the small Teres minor and one of the worst ones in youth pitchers is “LL shoulder” where the humerus levers over and against the pech over stressing and sometimes breaking the very large proximal humeral growth plate.
I have heard an MLB commentator say that a pitcher is "throwing across his body"
This would be why you should not be listening to non-credentialed so called experts discuss mechanics when they have proven to be in the past and still continue with their “yard myth” mechanical explanations of how thing work that keep the injury rates at a high level.
“He said this is a good thing because the ball will fly in an angle to the plate instead of straight to the hitter”
I think proper force application resulting in proper recovery positions that also allow you to be in the best position to defend and avoid line drives back at your dome are optimal. By staying tall, attaining a high humeral vector during ball drive and throwing your hand directly at the target by pronating your drive and release, rotating your shoulders 180 degrees with your arm (Humerus) staying in alignment with your shoulders.
Absolutely not, this is not normally taught anyways and has always been put in the realm of seemingly unimportant and non-injurious when the opposite is actually true.
“Is this an important quality in pitching?”
By performing the standard traditional over early rotated shoulder rotations from the leg lifted gateway the shoulders then never finish fully rotated at the end causing this “across the body “ finish from the shoulders only attaining about 90 to 110 degrees of rotational finish causing this poor postural finish.
“the first left turn circuit”