I use .44 mag handgun, at bowhunting ranges, and broadside. 240gr semi-jacket soft point. Plain old Winchester white box or Remington yellow & green box.
Both the bullet and the arrow will punch straight through both lungs. Even though one cuts and the other crushes, I don't see a big difference between them.
I've never used a 9mm, but I would if given the opportunity. Gold Dot 124gr from Glock 26. I carry it as my CCW. It is on me when I hunt with a rifle or shotgun. Left my shotgun at the bottom of a tree once to climb up and fix the treestand area above. A giant buck came through and you bet I was digging under my hunting clothes for my Glock. I would have taken a 10 yard heart lung shot on him. But he jumped off as I drew. No firing at a running away deer.
So in my opinion a 9mm defensive round on a standing broadside deer is fine. Under 20 yards sounds right for 9mm. You want precision and power.
Going up in caliber gives you more distance and more power for more difficult angles. That's why I like .44mag for handgun and .308/.30-06 for rifle