TAB, not a simple question.
1) Which 1911.
2) What caliber, remember you can have a Grizzly .50 magnum, 10mm, 45 super, .50 GI, all capable of hard heavy hits on light skinned animals, within a given distance. You can also have a 1911 in .22lr, .38spl, 9mm, .38super etc... not normally recognized as deer cartridges.
3) Why? Well maybe thats all you have, maybe you like to challenge yourself and limit yourself to more basic or inferior tools like a bowhunter, you know the limitations of yourself and your equipment and will only work within those parameters.
To answer your question:
1) YES, If the hunter is willing to work within the limitations of the cartridge and the accuracy of the particular pistol and her/his personal capabilities.
2) Best tool, no.
3) Adequate tool yes, within its limitations, remember the .45acp is meant to stop things in the 250lb range by design.
Your picture of the donkey in the well is disturbing, I hope you got it out alright. The skin looks so very loose it could be dead or very emaciated.
As for shooting ourselves in the foot, 1) RULE #1, NEVER POINT A GUN AT ANYTHING YOUR NOT COMPLETELY WILLING TO DESTROY.