The .270 is much more "devastating" -- than the .223 and is really not a comparison--much longer and harder hitting. The .308 is comparable to the ..270 in the real world up to 300 yds or so -- and arguably a better choice in this range for elk with more ft. lbs of delivered energy -- but otherwise beyond this yardage is where the .270 takes over. .270 flatter trajectory at higher fps than the .308 or .30-06 given similar bullets or let's say certainly comparing each round's most effective "all around bullet" - which would = 130 for the .270 (not much bullet choice in teh 130 territory for the .30's) and 150 gr in the .30 calibers. The 130 gr is where the .270 built its reputation and still shines though 140 gr has made some true believers in the last few years and is hard to argue against.
The .308 is much more comparable overall to the .270 than the .223 and is mostly a matter of choice between the 308and the .270 in the short to medium ranges. Long range -- 300-475/500 yd -- western hunting for mule deer and bighorn sheep is where the .270's reputation lives on--the quintessential western deer gun - "The Rifleman's Rifle" (Model 70 Win more specifically, early on) as the media has put it for many years. Shorten the range up just a bit and you're talking similar results with the .308 on that sized game...and preferred by some hunters for its (the .308's) short action = a handier "mountain rifle" right out of the box in most barrel lengths in which it is popular. It is true that the 7mm Mag will do it that much "better" than the .270 yet re range, trajectory with more lbs ft at the receiving end, but at a "cost" of wasted blast and recoil...for thin skinned medium sized game not necessary. The .280 Rem is a near identical twin to the .270--so much so that it's barely (some would say unfairly) been noticed in the marketplace over the years, coming onto the scene more than a 1/2 century later than the .270. The .280 is seen by many as having the advantage of a better - and slightly heavier - array of bullets to choose from and therefore a slightly better choice for elk duty--if it comes to that kind of decision.
It's almost like the .270 is an 80/20 (deer/elk %) +/- 5% gun--meaning when purchasing the gun there is at most a 20% "stated" chance it will also be used for elk, while the .280 roughly 70/30, the .308 a 60/40 (within its range as any of these) and '06 50/50. Neither the .270 or .280 are considered elk specialists, though many have been successfully taken with them (shot placement is nearly everything)--or as good as the .30-06 (which isn't either, but is closer to "regular" duty/acceptability) for elk. As for the 7mm Mag is in the 55/45 zone as it has been said that, while it will outrange the '06 or any of the rest of the calibers noted here, it doesn't have what it takes to consistently (ideally) bust shoulder bone--for instance--when it needs to vs even a non magnum .30 cal. shot within its range. Any time you get over the 50% elk category, you should probably be looking at one of the magnums like the .300 and .338 Win Mags, or similar RUMs. And if you are beyond the '06's accurate/effective range, you should be looking at one of these anyway for elk.
Other than its well known role with the US military (M-16 and its derivatives), the .223 is seen more of a small game, varminter, survival/home/self defense gun, and while it will certainly kill a deer it is not preferred (and been known to damage some meat in the process). To me, to take a .223 deer hunting is being "under gunned." and some areas may not permit it. Ammo availability of all three calibers is very high. .223 is pretty cheap and .308 very plentiful (due to its military applications over the years). The .270 Win is only slightly behind the '06 amd .30-30 in its wide availability in most sporting goods stores. The .280 less so for reasons mentioned--"fair" or not.
* = the essential 5)