The bottom line is this. An expert marksman who can pick his shots and will pass on anything less than Ideal can take elk with all of the legal cartridges for Elk. Also probably some that are not legal as well. A warden that I know has taken elk with the 243 and 250 Savage several times.
This said, Game all over the world has fallen to cartridges that were less than ideal including a lot of African game with the 303 British. This does not make using them a good idea by any means. If the 243 is Legal where you hunt and you want to use it, the final choice is yours. Opinions here are right on the mark though. At best, the 243 is less than ideal for sure.
I would put the 25-06 as the minumum and recommend more bullet yet. Just my opinion. I would add that I would use the 6.5x55 in a skinny second.