Why mess around with a gun that MIGHT kill a deer when you can buy a 243, 6mm, etc, with little more recoil than a 410? Either of these two rifles with a front shoulder hit will penetrate vitals 90% of time, not 10% of the time a 410 will. My first gun was a 410 when I was 10. It's a real nice bird, small mammal, starter gun, just not a reliable deer gun.
The main reason to use a .410 in this situation is because that is what the OP has and asked about.
Regarding reliability, it is still where you hit them...bottom line.
I've killed about 150 whitetails with all manner of implements and I don't shoot them in the shoulder with anything, due to meat loss and the fact that the main mass of the vitals is a bit further back. The heart is tight behind the foreleg, and 1/4 to 1/3rd of the way up, surrounded by lungs. Clip the back edge of the front leg a little low of center and you're in the money.
I see these choreographed outdoor shows where they shoot the deer in the shoulder or high in the back to make a dramatic one shot knockdown for the camera, but they don't show the often-required follow up shot on a squirming animal, nor the bone chips and bullet frags in the 5 or 8 pounds of wasted meat, often backstraps, that went into the trash.
Deer are not at all hard to kill. A 40# bow will send a sharp arrow right through one and kill it dead. Killing 50 deer in a two year period on crop damage permits taught us a lot. We saw that a .22 short CB will shoot straight through a deer's rib cage area and penetrate the heart & lungs. The resulting reaction was very similar to an archery hit animal, but less spooked and quicker to go down.
A .410 slug is 1/4 ounce, or approximately 110 grains. At 1,800 fps muzzle velocity, muzzle energy is 790 fp, and at 50 yards, it's still 650 fp.
A .357 Mag, which most states allow for deer, and is approved by most people who handgun hunt, only develops 1,500 fps with the same weight bullet, for a muzzle energy of 550 fp and 450 at 50 yards.
One could realistically say that the .410 delivers more energy at 50 yards than a .357 at the muzzle....20% more.
But even after discussing that, I've gotta continue to point out that power is not the issue at all, its where we plant the bullet, or arrow.
The biggest limitation with smaller calibers is that you must be more particular in shot placement and just shooting them in the shoulder is not usually ideal with anything.
One of my students hunts elk with both a .260 Remington and with his .300 WinMag. The .300 WinMag's additional penetration capability affords him a lot of shot flexibility, and this is the rifle he chooses for his trophy hunts. He doesn't have to pass up many "reasonable" shots and can shoot one quartering away, for example, or even through the shoulders, and can still expect good results. He uses his .260 for shooting culls at his friend's ranch. He is extremely careful with his shots with such a small bullet, and would decline a quartering away animal or just a shoulder shot, but when he puts it through the ribs, the bullets make it all the way through and kill the elk dead as anything.
Were we given free choice, a small caliber rifle with decent sights or a scope would be an easier tool to use, and certainly affords us a little more shot placement flexibility, but we're not..,
If the OP's child can plant that .410 slug with certainty, it will surely kill a deer. As I had suggested before, the test is how that particular gun shoots slugs and whether it can be depended upon to place them correctly.
BTW, I got my first gun at 10 also, and it was a .410 as well, a double barrel Savage 311. Great little gun, still have it. Hunted almost everything with it, but never shot a deer.