The .22 Mag (or .22 WMR) is the ideal cartridge for harvesting elk, providing it was born today and you have a good reason for harvesting a freshly born elk with spots, (I can't think of any).
Now bear in mind when an elk starts out, it only weighs about 20lbs and it isn't very mobile. You can get pretty close to a newborn elk - so a .22 Mag is all you really need. Now an Elk is going to start growing pretty fast, and you will need a larger caliber at some point.
Fast forward 2 yrs and this elk has now lost his spots and weighs over 500lbs. The .22 mag is no longer the ideal cartridge unless you are hunting him inside a small pen. At this point a free roaming elk is pretty adept at keeping some space between itself and hunters and its body has gotten massive.
A tiny .223 remington would certainly have more potential at this point, especially with good bullet choice, but its not going to really open up many shooting opportunities. The .22 bullet is just too light to penetrate 300yds of air, an inch of hair, a 1/4" hide, several inches of meat, a bone, and then a vital organ. Sure you could dump an elk at 30yds with a headshot, but that baseball size brain has a lot protecting it, and a deflection is very likely if you aren't perfect. Lets say you have access to a ranch that has elk eating off a haystack and they aren't afraid of the ranchers truck, could you double lung an elk and then watch it die within a few hundred yards? Probably, but you're getting close to not reliably having the horsepower to damage both lungs, so real world options with this tiny gun are very limited.
Enter the light rounds of .243, .257, 6mm, 6.5 creed etc. This class of guns can reliably double lung elk out to a considerable distance 3-400yds lets say. They can also take out a heart or Liver at closer distances, but one must be very careful to not hit heavy bone or take on too much angle. This is a pretty ideal cartridge for those willing to only take well presented broadside shots out to about 300yds. If you are an experienced and expert marksman, this distance could be stretched out to maybe 400yds reliably. You've also got the option of neck shots in the tight timber, where torso shots may be hard to discern. This Class of cartridge is low on recoil, therefore high on accuracy. A good hunter could feasibly get an elk every season with a rifle this size, but self discipline and experience is a must!
Stepping up to medium rounds. Enter the 270, 7mm .308, .30-06 etc. class of cartridge. This is a pretty good choice as you can typically get a bullet into the vitals at most angles and do this out to 400yds or so. A good marksman can penetrate into the vitals of a broadside elk at much greater distances with these heavier bullets. Keep in mind that Elk hunting public ground is no easy task, these are big skittish animals that don't always present ideal shooting conditions, and seasons are short. A cartridge in this class is going to afford you a few more possibilities while still keeping recoil and muzzle blast to a manageable amount. There is good reason these calibers have been so popular with elk hunters. You don't need to be a sniper or expert in anatomy to take a shot at an elk with a medium caliber gun.
The Magnums... Is bigger better? .300 win .338 Lapua .375 H&H etc. An elk hunter may spend many days and hike a hundred miles before presented with a shot at an elk on public ground. When that moment finally arrives it may be a bull elk running directly away in thick timber, or it may be 700yds across a canyon, do you have enough gun? A .338 RUM with the right bullet can do things a 30-06 can't. Should you take a Texas heart shot on an elk? That's up for debate as some would say wasting one quarter to put three quarters in the freezer is worth it. Others say to pass on a wasteful shot. I think it depends on the frequency a person is afforded shot opportunities, but we can all agree that shooting an elk up the rump with a 6.5 creedmore isn't going to work out. Extreme shot opportunities come at the cost of dreadful recoil and muzzle blast, enough to cause the toughest of men to flinch. I've seen many elk missed entirely, or hit poorly because a person used too much rifle. I've also seen many elk get harvested by magnum rifles that couldn't have been taken with lesser guns.
I've missed an elk with a Large magnum round that could have easily been headshot with a .223 remington. It was 30yds away and bedded down with its head resting on its rump facing me. It was such a "gimmie" shot with my .338 RUM that I put my crosshairs on its nose, (flinched) and pulled the trigger. I grazed the top of its skull and momentarily knocked it out. I wasn't even holding my gun when it woke up and escaped! I have no doubt that I would have taken the care to place that shot perfectly with a smaller gun. It wasn't the first time I've "jumped" on a trigger knowing my magnum could kill an elk without precise bullet placement. I've learned to shoot it better, and I would be hesitant to shoot a smaller gun, although I'm not convinced the gun I use is 'Ideal'.
What is your experience in finding the Ideal elk caliber?