For now, my own research into this topic has provided an answer I am ok with.
Yes it does.
After playing through the whole game, I decided to test this wherever I could. After doing a couple tests with really heavy Pokemon and the worse versions of the Heavy Balls, (Heavy Ball and Leaden Ball,) I can say that it does impact the catch rate by a lot.
For example, I caught Palkia at half health with one Leaden Ball, (small bounce too,) as well as an overworld Alpha Rhyperior with no food and no back strike, (didn't see the bounce on that one.) These are just the most recent cases.
If somebody can show me evidence one way or another that impacts my answer/question I will gladly take it, but this answer is good enough for me