Certainly I would. If it is wounded it will probably die a long painful death by infection. It it is a gut shot the probability of death is 100%.
Two years ago I had a big buck in my front yard. He “was down” but alive. He did not run even though we approached closely. I think he had been hit by a vehicle. He was put down cleanly and painlessly with a shot behind the ear. His meat was harvested and eaten.
The number one command in my lexicon is not to take the shot unless you know for damn sure it is a clean kill shot. If by error of marksmanship, it is your duty to track down the animal even if it takes you all day and kill it mercifully.
A couple of friends of mine hunt on my land. They are bow hunters and rifle hunters. As bow hunters they are extremely skilled. If not skilled they would not be allowed to hunt on my land. The greatest distance a deer has run from a bow kill on my land is 200 yards. When taken with a rifle it is behind the ear and they drop dead on the spot.
I despise non ethical hunters. They give the good guys a bad name. We do not like them!
I forgot to mention the Texas Parks and Wildlife Department. They do an incredible job in relationship to wildlife management. Each year they take several surveys relative to the wildlife and available food dependent on weather and resultant vegetation and food for the wildlife.
They then determine how much game must be taken to maintain the wildlife at its maximum best. When deer season is over they then do a second survey. If additional deer need to be taken they issue extra tags so more deer can be taken to maintain herd at the maximum best.
They and the hunters are wildlife’s best friend. They in conjunction with the hunters maintain the wildlife at its best. They and the ethical hunters are the good guys.
ps
I have a business in Salado, Texas. The town is overrun with an extreme excess of deer. The idiots that run Salado will not allow them to be taken by any method. They reek havoc on vegetation and deer car wrecks are frequent. The deer are relative healthy as we have had 3 years of good rain. When we have the next drought they will starve in mass.
My home is 3 miles West of Salado. We hunt here. Although the deer population in Salado is relative healthy our deer population is less but they are prime deer in great health. When we had three years of drought in the past our deer although not prime did not starve. Those in Salado did. Today Salado has more deer than ever in the past. The next drought will be very ugly for Salado deer.
I really hate stupidity guided by non logical emotion.