I've killed a pretty fair number of Virginia whitetails, all with buckshot. The first one, when I was young, I shot seven times with 3" magnum loads of number 1 buck. I later learned that the range was about 50 yards. Much too far, even though she went down every time I shot.
After that, after learning to judge range better, I never used more than two shots of the same 3" loads of Remington No 1 buck. The first shot put the animal on the ground. The next one insured they were never going to get up again (a couple I finished off with birdshot to end the suffering). Longest range, excepting that first one, was about thirty-five yards. The closest was about ten yards.
I also participated in skinning and gutting a bunch more buckshot killed deer. 000 buck breaks bones like matchsticks. I never used it much because my gun shot No 1 better, but the fellows in my club who used 000 swore by it. Their results seemed to back up their claims. The only time I ever shot at a deer using 000 I missed at ten yards when a tree jumped between the deer and me.
You could have covered that pattern with a coffee cup.
I never saw a deer killed with either 00 or No 4 that I know of. As far as I know, everyone in my club used either No 1, or 000, so I have no opinion about them.
I saw one deer killed with a slug (against the rules at our club). That shot was right through the heart, and the blood trail looked like it had been poured out of a bucket. That animal didn't go twenty yards.