Here we go with the old "you need a bigger bullet" and "you need to use those $2 each bullets" line of advice.
My last two elk were taken with:
6MM Remington using an 85 grain sierra bullet;
.25-06 Remington using a 120 grain Sierra bullet.
You don't need a super magnum, and you don't need exotic and erotic bullets.
What you do need is to know your limitations as a shooter.
Yes, a shooter needs to know their limitations - and that of their chosen cartridge.
While it is possible to take an elk with a .22 rimfire, that doesnt make such a cartridge a good choice for that purpose. In my 20+ years of elk hunting I have seen more elk wounded by a .243/6mm than any other caliber, so no, I dont recommend them either. IMHO the bullets are simply too light for elk - even when using Partition or TSX bullets. (AccuBond, InterBond, A-Frame, Trophy Bonded and North Fork are unavailable in .243/6mm.) One problem with a .243 is that, while it has plenty of reach, energy levels fall off fairly quickly. The Sierra 85g SPT, when launched at 3250fps, doesnt even make it to the 100 yard line before energy falls below 1500fpe, and the HPBT is even worse. A .30-30 with a 170g bullet handloaded to 2250fps does better, gives a bigger hole and will usually penetrate further.
A .25-06, while considerably better than a .243/6mm, is still a minimal choice IMHO. At least it can deliver 1500fpe out to ranges beyond which most elk are shot - a Sierra 120g HPBT leaving the barrel at 3000fps will just make it to the 250 yard line with that much energy.
By contrast, a .270 Win launching a 150g Partition at 3000fps delivers 1500fpe out top 480 yards. Not a magnum, not a $2 bullet, but a very effective combination and a far superior choice when compared to the .243 or .25-06 especially for those for whom elk hunting is a rare and maybe once-in-a-lifetime opportunity.
2$ bullets? What are you shooting, factory ammo? A quick check shows the following costs for .308 165-168g bullets:
$0.27 InterBond
$0.34 AccuBond
$0.42 Partition
$0.56 TSX
$0.74 Scirocco and A-Frame
$0.84 North Fork
$1.06 Trophy Bonded
In the .300 Win Mag the 168g TSX costs me $14.90 per loaded box of 20 and the North Forks run $20.50 for the same quantity. Cheap factory ammo usually costs more. The plain truth is that the money I could save by loading and shooting standard cup and core bullets at big game might allow me, after 4 or 5 years, to buy a cheap glass of wine with my dinner. Or, after a lifetime of hunting, I might be able to retire 10 seconds earlier.
Is there a difference in performance between small-diameter light bullets and fatter, heavier ones? Yes. Is there a performance difference between premium bullets and standard cup and core bullets? Yes. Are small caliber, light, standard bullets the best choice for elk? No.