There is nothing wrong with break open actions.
If there was shotguns wouldn't not have used them.
The difference however is that in basic physics when you fire it a break open action is always trying to push itself open on it's own pivot point.
Over the years the locking mechnisms have got stronger to the point where rifles like the T/C rifles are quite capable of magnum pressures but as a basic design they aren't as strong as a bolt action.
Falling blocks, more specifically those like the Farquarson/Ruger No.1 or the Browning have a breech block that drops down and is therefore perpendicular to the direction of recoil.
As such it is near well impossible to push the action open when the cartridge pushes back on the breech.
In terms of advantages break-open guns can be collapsed easily and many have interchangable barrels.
Most modern-break opens also are very accurate too and as I said before some are quite up to the most extreme cartridges.
For those who like light portable rifles they can also be made lighter and shorter because the action itself is very compact.
They do tend to like rimmed cartridges, hence the reason why many European companies chamber their break-open single shots, doubles and drillings in rimmed cartridges but T/C, H&R/NEF and many others chamber rimless without any problems and most European break-opens are available in 30-06 too.
I guess the advantage of falling blocks like the Ruger No.1 is that they can be loaded within an inch of their life.
They're super strong, often very accurate and are happy with rimmed or rimless cartridges.
In the long run a break-open gun can cost you as little or as much as you wish.
There's certainly no reason to avoid buying one for fear of poor accuracy or lack of strength.
Just ask the guys who own copious H&R/NEF rifles.