Back in the day I used to help cull a large number of red hinds. We used ballistic tip varmint type bullets and head shots to maximise return on meat (in those days venison was valuable). The head would pretty much explode with eyeballs all being forced out etc. I wasn’t pretty but very effective. We were using 25-06’s and 6mm wild cats with fixed 10x42 scopes and 26” barrels. Very flat shooting and never considered having to aim off. We would get as close we could before starting to shoot. Just made the whole process so much easier. Often we took 15 to 20 hinds of between 3 of us in a day. No moderators either, range finders, digital optics, twiddly turrets etc.
A ballistic tip varmint type bullet for such shooting causes massive damage - much more so than a more typical hunting bullet. Big downside that when on the odd occasion you had to take a body shot they wouldn’t penetrate the shoulder but would make a real mess of the carcass.
On game ranches in Africa, when culling for meat, tool of choice is a 223 with similar bullets and head shots.
Where head shots go wrong is when ranges are long and / or you are in very broken ground or woodland / scrub land. If the head shot doesn’t kill it can leave a highly mobile but badly wounded animal.
On open ground, once you have shot, wait ten minutes where you are. You should be in a very steady position. If any thing is going to try and get up, it will do so within a few minutes as it regains consciousness - you then have the ability to shoot it again. Nae chance if you are in lots of cover.
If you start at closer ranges you also have plenty of ability to shoot something again if it runs after first shot. If you open the batting at 300 odd yards you have little chance for follow up.
In those days my shooting ability, eyesight and general fitness was a lot better than it is today.