One thing about these discussions is that the question is, "What is the minimum caliber you'd use on an elk?" when it should be, "What's the minimum caliber you'd use on a bull elk?"
Too often we see people say "I've shot lots of elk with a .243" and they show a picture of a cow. Get close enough and shoot accurately and a .243 is probably fine for a cow. But I'm not going to try and kill a trophy 6x6 with one.
Cows and bulls are often two very different animals when it comes to size. When my oldest went on a cow elk hunt two years, I didn't hesitate to have him use a 6.5 Grendel. Inside of 250 yards it was plenty of gun. If that had been a bull hunt, I would've said .270 minimum.