Really odd that the coyotes didn't get him! Two things accomplished, you have a full freezer and the animal is not suffering anymore. Both pluses in my book.
My firend has a similar experience with a buck this year. He was looking for turkey and carrying his .17 instead of his bow (archery was still in) when he jumped a buck at 20 yds. He said the buck could barely stand and sort of hobbled about 50 yds and laid back down. That was late on a Saturday afternoon. On Monday (can't hunt Sunday in PA) he snuck back in there with bow in hand and again jumped this buck which bolted away this time - never to be seen again. My friend figured that this buck was in a fight and got the snot knocked out of him and needed a little time to recover.
But I agree, whitetails are a tough animal.