If on the hill and for red or sika only and where shots under 100m are uncommon, I would choose a .300 Magnum. For mixed woodland and hill with more 100-200 yard shots I would use a 30-06. I find the .30s hit harder than the 6.5s which with the right bullet work but seem less definitive. I haven’t used the “mythical” 270 so can’t say how much better or worse it may be.
have you ever tried the 7mms?
I am currently reading "Bell of Africa" ,and his thoughts on choice of cartridge, and caliber size needed to do the job on even large animals, is quite refreshing.
He has a large emphasis on the weapons and ammos reliability in the conditions they were to be used, (back in the day that was probably a larger issue) but he above all focus on shot placement and bullet construction rather than if he is shooting a .256, .275 or 303.
Now he was a very good shot by all accounts, and in Africa typically shot within 30-40 yards (with iron sights though, i imagine) , so it's not an exact parallel to the modern european deer hunting most of us probably conduct, but he didnt only shoot elephants with brain shoots, he shot a lot of game in general, and he was a keen observer of his tools and methods effectiveness in the field, and on how he could improve things, so his observations are interesting.
I do wonder what he'd chose now, with the current tech available, if he was for example stalking in England, Denmark or on the hill in Scotland or in the alps.