I have a .375 in the Browning A-Bolt and two custom Winchester Model 70 .375's. I also have a .416 Rem Mag in a 700 out of the Remington Custom shop. All those are stainless steel. My other .416 is a production Model 70 in the Safari blue/wood configuration. I unfortunately sold my .458 twin to the .416 Model 70 Safari. The point here is that I have shot some of the best of the American makes that are being discussed. All of them are good, but I recommend the Model 70 with controlled round feed and its other excellent features. I do not have much experience with the CZ, but do know they are rough and really need some major attention from a custom gunsmith to be top notch and also the stainless option does not exist. Custom gun work is really a good idea on any rifle, especially one used for dangerous game.
I have never had a single problem with my .416 in the 700, shooting everything up through elephant, but I do prefer the Winchester action. Stainless steel is nice despite what others may say. Sometimes I am tired at the end of the day and don't get around to cleaning my rifle. All you have to do is look at a PH's rifle in a dry. southern African country to see that corrosion takes its toll even in the absence of wet, salty conditions. Our bodies add the moisture and salt, and blood does the same. I much prefer the looks of blued/wood rifles, but stainless/synthetic is more practical.
The Browning A-Bolt is a great rifle, but a month ago I had the magazine jam on me as I was trying to recharge it after firing all my rounds into a cape buffalo. My PH went ballistic and told me to leave my Browning in the US and bring one of the Model 70's next time. (The jammed magazine was not a problem then because the buffalo was on the ground after the third round and never moved again, but under different circumstances a jammed rifle will get you killed.) The magazine has been my only issue with the Browning. I like the 60 degree bolt rotation. The reason I took it on my last safari is that my son took his .338 in the exact same model A-Bolt and I wanted to be able to swap rifles without having different safety positions, bolt rotations, feel, etc. I have done some custom work on the rifle, but it is still not a Model 70.
The Model 70's are the best. My factory Safari rifles were great out of the box except for needing some trigger work, as almost all production rifles need. I can recommend Remington, Browning and Winchester, but if you have the chance to pick up a good Model 70 stainless, or even a Model 70 Safari, at price of $1,300 or less, buy it and feel lucky. The actions alone are going for $1,300 from the custom gunsmiths, if they even have one available. The NP3 and E-nickel finish from Robar are both excellent and should give you a lifetime of worry-free use unless you drop your rifle hard on sharp rocks multiple times. The black Teflon finish is the darling of the custom gunsmiths, but it WILL be damaged by hard use that would do little or no damage to the NP3 or E-nickel finishes. Any of these finishes can be removed and reapplied if they are damaged. I prefer to spend the extra $300 finding and buying a stainless rifle. Keep looking on Gunbroker.com and Guns America, and you will eventually see a Model 70 that will fit your needs.
If you are just going to shoot bears and such in North America, go buy a .338 Win Mag in the Model 70 Stainless and be happy. I would never own anything bigger except for Africa. The .338 Model 70 Classic in stainless is much easier to find and generally considerably less expensive, and you can usually get one new in the box for less than $1,000 and have the greatest all round caliber ever created outside of hunting African dangerous game. Have the barrel cut off to 23 or 24 inches and recrowned if it comes with the 26 inch barrel. That length is too long and unnecessary in the .338. The .340 Wearherby is a different story. I have three .338's, not including the Browning that I gave to my son. All mine are custom, stainless rifles from Rifles, Inc.: a very lightweight Remington 700 (6 pounds, 2 ounces scoped), a lightweight Model 70 (7 pounds, 2 ounces scoped) and a standard weight Model 70 (8 pounds, 3 ounces scoped). I do not need a muzzle brake with any of them, although the 700 came with a muzzle brake that I took off and replaced with a thread protector. I certainly understand if you just WANT a bigger cartridge. Most of us in this forum feel the same way.
Good hunting!
