You both have FINE elk rifles! Have no doubts.
And your choice of bullets/ammo is perfect. 150 to 175 grain bullets are the best and I favor the 160 and 175 grain bullets.
I used to have a 7X57 myself (used with both 160 and 175 mentioned above) and I know of several successful hunters using the 280. I have killed lots with a 270 Winchester.
I have hunting and guided elk hunters now for 44 years. I have seen a bunch killed with 25-06, 257 Roberts, 6.5X55 Swede and a few with 243s.
The thing I have learned in my many years of killing elk and seeing elk killed is that the bullet you use if FAR more important then the cartridge you fire it from and of course marksmanship is the most important thing there is.
As a guide I have had to track down a lot of wounded game in my life. The #1 cause of long tracking jobs is poor bullet placements
but the #2 cause is poor bullet performance. MANY bullet used by new elk hunters are chosen because of their accuracy which is the worst reason you can choose a bullet for big game. If you have 2 MOA you have all you truly need for elk out to 500 yards and 500 yards is a lot farther than you are likely to shoot and as far as you should shoot for ethical reasons.
But I have had to track about 30 elk in my life hit in the RIGHT PLACE with poor bullets and those bullets broke up and didn't penetrate well.
As a side line, I have been with many clients that made "less-than-perfect" hits on elk with good bullets and the tracking job was easy, or in some cases, the elk didn't go anywhere after such a hit,because the bullet went clear through and damaged the innards enough to cause then to lay down.
Hits far back in the lung area that go through the diaphragm and through the liver area through the guts and out the back with good bullets will still bring down an elk in a short distance and some times a very short distance. The same hit with a fragmenting bullet often results in an elk going between 400 yards, and the worst ones I have tracked went over a mile.
One of the very worst recovery jobs I ever did was a very big bull my client hit with a 7MM Mag. He shot it from a high angle above but the bullet hit the ridge of the shoulder blade. If it has held together it would have come out the bottom of the chest on the off-side. At the angel his hit was perfect, but his bullet was not! The bullet came completely apart and not one piece of lead of jacket made it into the lungs. As I was to find out gutting it many hours later) That shot was about 30 minutes after 1st shooting-light in the morning. I and my hunter caught up with that wounded bull just before last light and he killed it with a 2nd shot into the ribs I gutted it by flash light and we had to pack out the first pieces (back legs and hams) in the dark for about 4 hours, where there were LOTS of grizzlies around. Not any fun, I can assure you.
The bullet was a Sierra "game king" of 150 grains as I recall.
I made a very similar shot with a 44 mag handgun with a 4" barrel a few years later. I used a cast 265 grain SWC bullet. The elk staggered and fell in about 15 feet. The bullet broke the shoulder blade and exited mid body. A 44 mag is a powerful gun, but no where near as powerful as a 7MM Rem Mag. All that power was of no value however because the bullet didn't use it the right way
You and your son are JUST FINE. Use your 7MMs and use the heavy Nosler Partitions and you'll probably need a sharp knife and packs too. Your guns and ammo will put elk down and you'll need the knives and pack frames. I hope for some hard work packing in your near future.
Happy hunting
