There may be a few reasons to choose the .300 Winchester Magnum over the .338 Winchester Magnum but being "quite a bit flatter" isn't one of them. When using similar weight and configured bullets, the .338 is pretty much the practical equivalent of the .300 in terms of trajectory and, in some cases, can be seen to be the slightly flatter round. To cite an example, the Federal ammunition catalog reports the .300 Magnum with a 200 grain Trophy Bonded Bear Claw bullet, when sighted-in (zeroed) @ 100 yards, drops 4.2" by the time it reaches 200 yards and 15.2" @ 300 yards. The .338, using the identical but even heavier bullet (225 grains), when zeroed at the same distance of 100 yards, drops 4.1" @ 200 yards and 15.1" @ 300 yards.
There just isn't much practical difference in terms of trajectory between these two fine, proven rounds to base a decision on. As mentioned, other factors (cost, recoil, availability, etc.) seem more relevant when or if important to the individual buyer.
If one is going to compare two cartridges of different calibers, it helps to compare apples to apples. Comparing the .338 at the light end of its bullet weight to the 300WM at the heavy end, then saying there is not a meaningful difference in trajectory, is not really accurate.
My 200gr Accubond 300WM load is 2950fps; zeroes at 100 yards there is 9 inches of drop at 300 yards. Even with a 225gr bullet your .338 is dropping 15.1 inches; with a 250gr bullet it will likely be even more. That's a pretty significant difference.
Likewise, the comparison earlier claiming the 270Win and 300WM were in the same class using a 165gr bullet for the 300WM...they sell those, but you're really not using the cartridge to its potential until you get to 180gr. There is a substantial difference in performance in the field between the two, they are certainly not equivalent chamberings.
As for the OP, plenty of people hunt elk with the 270. I've seen video of a friend taking a moose at 400 yards with one...marginal but with a follow-up it got the job done (a 300WM would have done it in one). Define what big game you are hunting. If it is not bear, and you are not shooting game over 400 yards, your 270 is just fine. A 300WM will extend your range and take you into bear.
People have magnum-itus these days. A few years ago I saw a hunting show on TV that said the 300WM was the smallest acceptable cartridge for whitetail deer. I shoot my 300WM more than any other rifle but for deer within 400 yards I'd choose a smaller option. It's overkill.