HGH promotes bone density, muscle mass, energy, healing, etc. It's nicknamed the 'fountain of youth' for a reason. Part of the aging process has to do with the decrease in naturally occurring HGH. So supplemented HGH, especially for players who would otherwise begin experiencing deterioration in their playing skills, will find themselves stronger, leaner, faster, more energetic and able to heal much more quickly than their peers or they otherwise would at their age. That seems to me like it absolutely could result in more home runs.
The drug doesn't do any one thing but rather aids a ton of various bodily functions. That's why it can be argued it's as good or better than steroids, because it has more effects without being as noticeable.
HGH contributes a ton. Its overall impact could make a player much, much better. Healing quicker is a part of it, but it goes well beyond that.