The problem is between your ears. And partly in your ears. A 7 mag is a lot louder than 30-06 and the noise tricks your brain into thinking it kicks harder. A muzzle brake just makes them louder.
A 7mm Rem mag and a 30-06 shooting similar bullet weights from equal weight rifles have nearly identical recoil. Since the 7 mag has the word "Magnum" on the headstamp you've convinced yourself that it kicks hard. If you think it's going to kick hard, it will.
With either a 7 mag or 30-06 I'd avoid ultra-light rifles. But you should be able to keep the weight under 8 lbs including optics and mounts and have tolerable recoil. The old walnut stocked ADL's had no recoil pad. With modern recoil pads such as the Limbsaver and others felt recoil from will seem closer to 243.
You are correct about the issue being between the ears, but it is not due to the intensity of the noise. Bought muffs when that first used Colt AR-15 came home with me back when I was in my 20's, and have used them on every shooting session since. Still in the range bag today along with the new set of electronic noise cancellation muffs. At first the 7mag was a joy to shoot, I liked the recoil for some strange reason, and was punching excellent groups in paper and the ex-wife's 8 inch stuffed animals out to 400 yards. That was a long shot "back then". Reloaded buttloads of 140, 150, 160 and 175 gr Sierra, Nosler and Speer projectiles in Rem brass, over H870, IMR4895, H4895, IMR4831 with CCI mag primers. Loved the 140/150 Sierra PSP, Nosler Ballistic Tip/Solid Base, they were easier to shoot well and didn't eat up as much powder per shot as the heavier bullets. Seems it was almost 80grs of H870 powder every time the trigger was pulled on 160 &175 projectiles. However, no records survive from that ancient era.
Back to the point, I was thrilled with the 7mag, had supreme confidence in my ability and the rifle/load capabilities. Until I started missing....
Shots were going wild, groups began to grow and the variances were erratic. Was forced to sell the ADL due to hard times in the late '80s, still saddens me to this day. However, when I started collecting again I noticed the flinch, from .22lr rifles all the way to pistol shooting. My brain had been bashed by the 7mag to the point that there is a lunge at each trigger pull. I remember the embarrassing moment when the fellow on the lane next to me at an indoor range had to point out that the .45acp slugs from my Springfield XDS were bouncing off the floor! I've gained some semblance of control shooting a 10/22 with a hunting scope, AR with .22lr adapter and deep breathing, as well as taking time and breathing correctly on the pistol range. (Of course the 3.5lb Aero trigger in the AR556 might have helped a bit.

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Snapcaps brought the painful truth to light for all to witness. No matter what is being shot, AR, FAL, AKM, .243, 9mm, .45 or great grandmas .22lr single shot, I will flinch.
Several others have mentioned the Limbsaver recoil pad, so this will require some research. Looks like a lot of thumbs down on the muzzle break idea. Even with good electronic muffs?
Everyone's input is appreciated. Truth being, I miss the big 7mm, I still have dies, bass, bullets and powders to reload for it (not to mention loaded ammo) but maybe it is time to step back to a less intense 7mm. 7mm-08, 7x57 or something in that range. The only 7mm in the pack right now is a Rem XP-100 in 7mm BR. It may not be a 6.5 ShootzinBangin but it will punch one hole groups as far as I can shoot it.