I'm thinking it all comes down to bullet choice. I would not hunt elk with a 243 personally.
Between the other 2, my bullet choices would be:
.257=115gr TSX
7mm=120 TSX or heavier premium bullets of various makes like NPs, or Accubonds, and the like.
Both of above said 2 bullets can be pushed to about the same velocities out of the given cartridges. The 7mm having a very slight edge in BC and a noticeably larger frontal diameter.
For Varmints:
Just about any .257 bullets
7mm=100gr HP bullets from Hornady or Sierra, most likely
The 7-08 has infinitely more good bullet choices for elk than the 25-06.
The 25-06 has infinitely more good varmint bullet choices than the 7-08.
IMO, shooting an elk with not enough bullet is a no-no. But, shooting a prarie dog with a 50 BMG is fun.........