There are several muzzle loaders pistols that can do an admirable job on whitetails. Some of the members here are from wide open spaces and a close shot is reasonably unheard of. Here in Penn's woods the average shot at all deer is about 30 yds. Certainly not a problem with a well stoked muzzleloader pistol. The problem is that most of the "economical" items on the market, are poorly suited for precision shooting. Heavy trigger pulls, unbalanced construction, etc. The traditions trapper is heavy and clunky, and needs tuning as built, but is a sufficiently powerful pistol to take whitetail. A few hunters here in PA (where we have a flint only season for three weeks,) use 50 cal Pedersoli bounty hunters. Rather long and unwieldy but well capable in a practiced hand. Some of the more modern design pistols can achieve rifle ballistics easily. The pedersoli Kentucky pistol in 54 is also a fair whitetail gun.
Some of the internet black powder celebs have been preaching that round balls and side locks are insufficient for deer even in rifles. Some where some muzzle loader fans got the idea that whitetail require the equivalent of 19th century English elephant loads to bowl over a Bambi.
You do need more than an Old CVA 45 cal Philadelphia derringer. But there are models that can develop capable 50 yd whitetail ballistics. But you also have to do your part to be proficient. The state of Maryland permits cap and ball revs for whitetail, fallow deer and sika. If you can load 45 grains and a PRB in a 9 inch barrel, you have already beat the ballistics of the revolvers.