Cartridges I’ve used for elk include 25-06, 260, 270Win, 7mm-08, 280Rem and 30-06. Bullets were 115 gr. to 180 gr. of many types. All worked well. A .243 caliber bullet is the legal minimum where I mostly hunt. I will soon shoot a 6mmARC for elk with 103-108 gr. bullets most likely, it will also work well. Larger cartridges also work well, but not for me as I hate recoil. “Deer” rifles work well for elk when employed correctly. That means shooting no further than your capabilities or the bullets minimum expansion velocity, whichever is less.
The key is your ability to place an elk capable bullet in the right spot with adequate velocity. 270Win and 280Rem get the most use from me and my group. A 100 gr. 243Win bullet in the vitals is lethal fast, a 180 gr. bullet in the guts from a 300 mag of some sort you don’t shoot well is a problem. I have a buddy that shoots a 340 Weatherby well, I’m afraid of it. We both kill elk often.
An outfitter I know prefers new elk hunters use a 243Win because they can shoot it accurately. He keeps ranges moderate, usually under 250 yards, does mostly cow elk damage type hunts. He says most shoot once, the elk trots off a bit, then falls over. He has very few wounded elk tracking jobs.
To a degree it is a Ford, Chevy, Dodge, Toyota, Nissan, etc argument. Pick something you will shoot well and will shoot a lot from field positions. Another good plan is to shoot a “training” rifle often and your main elk rifle less. There are thousands of opinions on the matter, poke around some, and good luck.