I have not seen bore diameter make any difference in how quickly the deer drops. For a long time I was subscribing to the thought that my small, fast 6mm & 257" rifles were killing deer faster...
Then I hunted with a traditional 50 cal muzzle loader using a soft 385 gr hollow point for a couple of years. Two shots. Two instant kills. Hmm. From a 1300 fps big bore chunk of lead. Well crud, there went my light-fast bullet superiority theory.
Mostly I've found that when I do a good job of shooting, the critters either drop where they were, or there's only a very short tracking job.
I have to admit though, that mostly I hunt mule deer in fairly open country.
So no, I don't worry 'bout how big around the bullet is. The past few years I seem to bounce back and forth between hunting with my 25-06 or my 30-06, and the results all tend to be about the same. They all seem to work if I shoot well.
Regards, Guy
