Speaking from personal experience I would absolutely take an elk with my 6.5cm. I can not speak for the 22cm because I have not used it yet. Do I believe the 6.5cm is the perfect elk rifle, absolutely not! But with a good bullet, good shot angle, and moderate range I would not hesitate to take the opportunity. Elk are tough no doubt but they absolutely are not bullet proof. A smaller quality bullet in the boiler room will always garner more success than a poorly placed 308 or 338 bullet.
Speaking to some of the other things mentioned like 308 being too much recoil, this is absolutely a real issue. There are many folks out there like new shooters or those of us with severe shoulder injuries who truly do struggle to PROPERLY tame those rifles. My wife is an excellent example, she didn't get into hunting or shooting until she met me in her mid 20s. I started her on my 223 bolt gun then upped her to a 308 because it was what I had. Almost immediately she developed a flinch, adding a muzzle brake created even more issue due to the increased noise making her jump. I dropped her to a 6.5cm and it was like the stars aligned! She went from not being able to connect at 150 yards to being lethal at 400 confidently.
I know many guys think smaller cartridges aren't good for more than coyotes like yourself but I can absolutely attest to their lethality on larger game. The 6.5cm is my go to for anything smaller than a mule deer. It has replaced the 243 in recent years. The 6mm ARC has also become a favorite of mine for the same role. I know folks who regularly take elk with the 6.5cm, 25-06, and 257 Roberts. My dad and grandfather have both taken bull elk with the 243 while deer hunting in the past as well.