Should pulling be a part of your hand balancing practice?
Well like many other things the answer is subjective depending on what you are trying to achieve.
The first thing to note is that no amount of pulling work will have any direct benefit on your hand balancing.
In fact; being strong in pulling movements can actually hinder your handstand progress. This is because being a strong puller usually comes with other baggage such as tight lats and forearms which can make it more difficult to create a better line with which you can find a resting position in the handstand.
Hanging on the other hand can be useful for building shoulder mobility and decompressing the body.

The answer I give here is of a similar theme that I often give. It's about assessing your goals and figuring out what you are trying to achieve.
Are you training to solely specialize in hand balance? You don't really need to do any pulling, though I personally would not give the advice to avoid it completely.
Generalist physical culture enthusiast? By all means, experiment with a bit of everything, but know the demands and costs of the disciplines. I can progress in pulling with 1-2 sessions per week but would see very little progress if I did the same with hand balance. On the same note, heavy pulling sessions can have a negative impact on your handstand skill work because of how the body responds.
Climber, aerialist, pole dancer or other pulling dominant athlete? I think some basic handstand work is valuable in balancing out the shoulders and help prevent injury,

Personally I've always included some kind of pulling work in my physical practice, as pushing strength came easier to me. I've also never called myself a hand balancer; have performed as an aerialist, and enjoy the combination of pulling and pushing to balance out the upper body(aesthetically and functionally).
What are your thoughts on balancing out a handstand practice with pulling strength?