@Cervus elaphus, IMO you don't need a 30-06 with more weight. 7.5-8# (3.4-3.6 Kg) is more than enough to mitigate the recoil on this caliber with proper practice and a proper fitting rifle.
A proper fitting rifle is one of the things I've found most overlooked when buying a long gun. Rifles and shotguns are made to be general standard to fit an average sized male (whatever that means). I know it's not common or inexpensive, but having a rifle properly fitted to YOUR body makes a huge difference in felt recoil. It's worth the time, effort and money to have this done.
Second is practice. Consistent practice, including much more dry fire practice than range time. Range time is to confirm what you are doing in dry fire drills. This also saves on money and considering the current ammo situation, this is a good idea.
Last is technique. As caliber goes up, flaws in technique become more apparent. Poor technique shooting a 22lr, you simply can not do with a 30-06 without paying a price. Add to that, reduced rifle weight or larger caliber and the problems can compound quickly to a negative shooting experience. This can lead to flinching and the problems only get worse from there. The point it to make shooting fun and still test the limits without going over them. Proper positive shooting experience with a rifle builds confidence to be able to move up in caliber or down in weight for a given caliber.
It was not that long ago that I thought I needed a rifle to weigh over 10# in 375H&H. Through consistent practice and with a proper fitting rifle, I can easily shoot a 7.75# (3.5 Kg) rifle with ease. Here is a picture of the rifle and short clip of me shooting it. I'm 70" and 160#, by all accounts not a big guy and this was no problem for me. Did I shoot this all day? Of course not, that would exceed my limits. But only through practice have I learned this. You can do it too.
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