To be truthful, and for the most part, no.
I have never shot a 7mm-08, but the rifle that I have probably killed the most game animals with is my .257 Ackley which is somewhat similar to the 7mm-08. I built that rifle primarily to use on medium size big game such as deer and pronghorn antelope, and have been lucky enough to have also shot a few bighorn rams and a Dall ram with it.
I also used that .257 Ackley on a DIY, solo, backpack bighorn sheep hunt in SW Montana where I ended up killing my second best 6 pt bull elk with one DRT shot. And on my Dall sheep hunt I also used this rifle to make a 250 yd one shot kill on a Mountain caribou bull. Both of those bullets were 117 gr Sierra GameKings, and the caribou shot was a complete pass through and we heard the bullet ricocheting off the rocks beyond the bull.
I have several friends that do most of their Montana big game hunting with their .22-250s or .220 Swifts. This includes deer, antelope, black bears, quite a few elk, mountain goats, and several buffalo. These guys are all excellent shots, have been hunting all of their lives, and, being residents, have all the time in the world to make their shots. They also know their limitations, and don't take extreme range shots, or shots at poor angles.
There is a lot written on these forums about using a big enough gun in case the only shot opportunity that you get is at either an extreme range or is a poor angle. And, many of these hunts are high dollar, guided hunts.
Not all shots result in one shot, DRT kills, and we all have had animals run off after being hit. Tracking and finding a wounded animal can take hours and even days which on a guided hunt we may either not have extra days to find a wounded animal, or we would rather spend our days hunting, perhaps multiple animals, than tracking a wounded animal.
This is why many recommend hunting with larger calibers, or larger cartridges.
I started hunting with a .30-06 for everything from gophers to elk, and matching the bullets to the animals. As my hunting progressed, I bought more rifles and began matching the rifle to the animal.
I have killed mule deer and pronghorn antelope with .22 caliber rifles, and I have killed 30 lb African steenboks with a .375 Rem Ultra mag. Neither caliber would be my first choice for those animals, but sometime you have to use the rifle that you have.
I have confidence in my shooting ability and in my rifles. For the most part I try to match the rifle, the cartridge, and the bullet to the game that I am hunting. If I am going on a multi animal hunt, I will match to the largest animal.
I also believe it is better to err on the large side, if you can handle the larger caliber.