If you want monolithic copper bullets to deliver the same ratio of quick kills as traditional cup and core bullets, on light skinned, small to medium deer, you're going to need to shoot them in the head.
Shared experience and first hand facts suggest the reality of a lead free hunting future is clear. Delayed expansion, over penetration, zero fragmentation, very narrow wound channels and limited or no peripheral wounding, and a relatively low percentage of impact energy imparted inside the vital zone... all this adds up to demanding you make absolutely no error whatsoever in your shot placement.
If you have a good dog, and/or are used to hunting light to medium game with harder bullets like partitioned, bonded or some of the tougher mechanically locked bullets, then not much will change. You'll be used to over penetration and a higher percentage of long runners and retrieving your deer potentially quite far from where you shot it.
If you don't have a good dog, and/or you need to anchor your game very close to where it was when it was hit, then you're going to need to up your game when it comes to deer anatomy and shot placement. Anything behind the shoulder, broadside, is a big risk. Heart shots? No, don't go there. Breaking the humerus or nailing an internal CNS pathway is pretty much a prerequisite if you don't want the deer to run. Outside of deliberate head and neck shots though, you're gambling, to a greater or lesser extent.
I respect the opinions of those that hunt or have hunted in the past with copper monolithics, I have no quarrel with you, each to their own. Its good to see there are some pragmatic opinions expressed in this thread. I am happy to be hung, drawn and quartered publicly by those that have a difference of opinion based on the experience of multiple deer! It's a game of percentages, we all get runners from time-to-time for whatever reason, but if you adopt monolithics you have to accept some of your metrics will change, and your habits will likely need to be adjusted accordingly.