The .303 is often compared with the .30-40 Krag, but its ballistics are closer to .308. The 174 gr Sierra boattail is what I use, I never liked the 200 gr although if I were going after bear I might change my mind.
The .308 develops a little more more velocity with the same weight bullet. I can get 2500 out of a 174 in a .303, but that is really pushing it and accuracy suffers, while the .308 will do 2800 easy.
I think where you really notice the higher case capacity and bigger energy potential in the .30'06 is in the heavier weight bullets. The .308 is very unhappy with 200 gr and Hodgdon doesn't even recommend a load for the 220, but the .30'06 will still heave a 220 out there at 2400 fps.
I'd have to route the string through/around something, I still would not want to stand behind it... (no wall/berm behind the benches at the outdoor range)
As for .30-06 loads, my favorite is the 180 gr for deer. My friends gave me a hard time at first "Isn't that a bit MUCH for whitetail??", but I really don't like trying to track a critter in shortgrass prairie in the dark, and most of my best shots are near dusk.
The 180 puts them down, now.
Besides, if you place your shot well, you don't mess up any more meat than you would have with the 150 gr. (which is not much, just a couple of ribs)
I have some of the old Herter's Round nosed ammo for the '06, much heavier than I'd use for anything but bear or maybe moose/elk, and admittedly do not recall offhand exactly what it is, I just put it up. I have not had call to use it, but it will keep...
I never loaded the .308 to the heavy side, with the '06 around I just did not see the need, and so far, I mostly use just it punching paper.
That 2800 is hotter than factory, what load are you using?