I am not going to get into what is better or "Best Quality", but, I will say from experience that the sidelocks are weaker at the head of the stock because of wood removed for the sideplates and the internal workings. Many sidelocks have very little wood bearing against the metal at the back of the action, so the makers use the sideplates and the tangs to help absorb some of the recoil;. That sounds feasable and it does work, until the head of the stock gets oil soaked from someone putting too much oil on their sidelock mechanisms. Most sidelock plates have a generally tapered profile that gets narrower towards the rear and then terminates in a rounded rear end. After the head of the stock gets weakened from oil or too many hot loads fired, the lockplates start taking more and more of the recoil till the stock splits behind the locks because they are acting as wedges to force the wood apart. To hold up well, sidelocks have to have wood that is denser at the head end of the stock than boxlocks.
Boxlocks, on the other hand, can have more wood left in them at the head of the stock. They are easier to get inletted well than the sidelocks, and as was mentioned by somewone else, they don't get as wet as a sidelock does. One of the main reasons that I feel Americans like sidelock guns is so they can take the locks off and play with them! More damage has been done by hamfisted nitwits playing around with their sidelocks than any other cause. For shear strength and perfect sear construction, the Lefever sideplated shotguns are about the best design, but nobody has bothered to make rifles with that design, yet. I also don't buy the old addage that the sidelock guns are easier to get a good trigger pull out of. All mechanical triggers can be adjusted correcly if they have the proper geometry to start with.
I say all this as a maker of many gunstocks, even though I collect and shoot L.C. Smith shotguns.
Here is a picture to illustrate what I'm talking about, most split right at the back of the lockplates, though. Bob H.