I'd get a short action rifle in 260 or 7-08 and shoot the 120-130 class of bullets. Both are mild recoiling. Any bullet in that class will kill deer and antelope with aplomb. Both calibers will run 2900-3000 FPS.
No need for a long action anything but it sounds like nostalgia may play a part in your decision.

"Flat shooting" is relative but typically is interpreted as fast, more powder and more recoil...typically.

Somebody will tell you the 270 is gay, which I believe it may be from what I read on here.