The 9.3x74r and 62 vs 375 H&H argument is pretty much the 308 vs 30-06 argument. Both work very well, but the 375 has the edge. Does it matter, probably not. The 9.3’s can be built nice handling trim rifles, and many doubles are built in small action size, and bolt actions are standard 30-06. The 375 H&H is of course a long magnum cartridge so needs a magum sized action.
Whilst plenty of doubles have been built for rimless cartridges such as the 375 H&H, 9.3x62, 30-06 etc, they rely on little spring loaded claws that fit the groove. There are ejector that rely on perfect timing to drop out the way to allow cartridges to fly out. These are generally fine, but .... there are always gremlins lurking. A double or any break action should really have a rimmed cartridge. You just need a simple extractor to lift the cartridge, and with rifle you don’t really need ejectors - a quick of the flicks, empties out and two more drop in.
Does this really matter. Probably not for a sportsman hunter. In most cases first shot does the job, and the second is back up. On driven piggies, two quick shots from a double is generally all you get - I would much rather two shots from a double and a quick reload, than three or four shots in a magazine and have to reload between each shot.
Looking at the piccies of the Browning over and under - they are generally good solid well built and reliable guns. Action will be the same as their over and under shotguns. Stock fit - well unless you handle it, difficult to tell, but this can be altered, and factor this in when bidding. There are now affordable options to get well fitting after market stocks if needs be for Brownings, Berretta etc
It is a double with open sights, and generally these are plenty good enough if you practice. But depends on your own eyes.
It looks like the rib has been drilled and tapped to add a weaver rail for a red dot sight. If you wanted to add a scope, you have options, but depends on the rib and how parallel it is to the bore. Cheapest would be drill and tap the rib and add a rail. Nicest would be to machine the rib and add swing mounts - but cost is c£1,000 to do that.
With a double, ammo is an issue. Most will only regulate with one type / brand / load. Add a scope you may well mess with regulation. Most new doubles were / are supplied with a good quantity of ammo that is suited to the rifle.
Once you have found a load that regulates lay down a store of these. The argument that if you loose your ammo, you can find ammo in the trading store in Petauke is really irrelevant. You need the ammo that regulates for the rifle. If you loose your ammo you are borrowing a rifle unless you are very lucky.
To some extent this is true for a bolt action as well, but with a scope it is usually a matter of at most a few clicks on the scope to get it back into action.
As to whether 9.3 or 375, I would suggest the 9.3s are much more like the everyday deer rifle that you shoot well. There is a bit more recoil but not a lot more. The last 9.3x62 I shot was a Blaser - and it did not really feel that much different to the 308 barrel that that rifle normally wears. Would I have noticed any difference in the field with a boar running past - no. On the same trip I shot 9.3x74 over and under - again not really any different to my 7x65R with a stout load.
With a 375 H&H it’s definitely a jump up from your normal rifle and to shoot them we’ll definitely takes practice and good recoil management technique. But a 375 H&H is not really a double rifle round, it is best in a good bolt action with a good scope and open sights, as it then serves for both big stuff in the thick bush, but you still reach out and shoot an impala at 200 plus across an open vlei.
9.3s and their ammo are pretty much priced like other rifles.
375s are a big jump up in price, ditto for ammo.
Find a rifle you like, that fits and then worry about calibre.