As requested, more grist for the mill.

American Boxlock Strengths and Liabilities

H&R - Strictly from a quality standpoint, the finest shotgun made in America, at the time. This was the only gun actually built under license from Westley Richards rather than designed to get around a patent.
Sadly, there are too few of them to be concerned with.

Parker - All hammerless:
Referred to by gun writers as America's "best" and "hellishly complex". From the standpoint of quality of fit, finish, materials and workmanship, these guns are second only to the H&R. I believe that
the Parker company's making of generally high-quality consumer goods directly influenced the quality of their guns. From a design standpoint, I would never refer to one as "hellishly" complex but they
certainly are needlessly complex. I wouldn't be surprised to find that the designer got paid by the part. The early guns used a complex, multi-piece linkage to translate the toplever motion to the single under bolt.
With all of the pivots and contact points in this linkage one would imagine that wear would "stack up" and result in much play in the mechanism. One would be correct. The latest guns and the Repro dispensed
with the linkage all together and operated the bolt directly by the toplever spindle. The cocking system remained unchanged throughout the guns production (and reproduction) run. This is a system in which
a spring-loaded hook in the barrel lump rotates a rocker arm in the frame that causes a slide in the bottom of the frame to rotate the hammers via hooks below the pivots, when the barrels are opened.
This design also requires an entirely separate mechanism to disengage the cocking hook when the forend is removed, this so that the barrels may be removed from the frame.
The cocking mechanism (and the need to house it) is the reason that small bore Parkers look out of proportion. Regardless of gauge, the action remains roughly constant in size.
It is in the ejector mechanism that I would use the "hellishly complex" description. An entire book/manual could be written about this mechanism. I'll spare you. A common problem with ejector Parkers
is that even though the gun is tight on face, the joint will be loose when open. This is caused by the forend iron wearing the forend lug at the point where they come into contact. This area experiences
accelerated wear due directly to the pressure of the ejector mainsprings. For all of its complexity, Parkers really do hold up and other than the aforementioned issue with the forend, failures seem
rather random. Most of the repair work that I perform on these guns is fixing the mistakes of others.
Lastly, the single trigger. Most of the American makers offered a proprietary single trigger and Parker was no exception. There were two versions, both mechanical-shift with an inertia safety block.
The late style was a fairly simple and reliable design and was used in the Repro guns.

Ithaca - Flues model:
a very simple design that was clearly optimized for mass-production with is round bolt (easier to drill a round hole than cut a square one), coil springs and simple cocking mechanism
that sees (basically) the barrel hook directly rotate the tumblers. This system is identical in principal, and similar in execution, to the way a Fox cocks its hammers. Unfortunately, accommodating
these internal parts didn't leave an excess of material in the frame. Flues models with cracked or broken frames are not a great rarity.

NID:
The ultimate Ithaca double with a rotary bolt (a la Fox and Smith), cam and pushrod cocking system (used in whole on the Model 21) and a whole lot of metal everywhere.
Clearly, this is a simple and strong design that should withstand much use and not a little abuse. The factory single trigger that was offered in these guns was actually a Miller trigger, built by Ithaca
under license. The Miller design stands out as maybe the simplest single trigger ever devised and they are trouble-free as long as they aren't messed with by incompetent fingers.

A.H. Fox - All:
I'll start at the top and work down. This design uses a rotary bolt, same as the NID (in fact, part-for-part, the design is identical) and the Smith. Where Smith decided to support each end of the toplever spindle
in separate parts of the gun, Fox (and Ithaca) supported the toplever spindle wholly in the frame of the gun. The result was much more rigid and compact. The Fox cocking system is simplicity itself and while
it shares the direct-acting principal with the Flues model, the execution is that used by W.W. Greener in their Facile Princeps box lock. The hammers are still directly lifted by the barrels but that part of the hammers
that actually engages the barrels has been "jogged" inward toward the centerline of the gun. This serves two purposes. First, it makes room for the coil mainsprings and the plungers and struts that drive the hammers.
Second, it allows the action bar to be rounded, again mimicking the Greener. Stepping out of the analytical and into the aesthetic for a moment, this is what I believe gives a Fox its superior "proportions" and shape.
The Fox uses coil springs throughout with the exception of the sear spring which is s leaf spring. Fox used a derivative of the Baker system for its ejectors, very simple and easy to time and repair.
Fox's "house" trigger was the Kautsky. THIS was the trigger that should have been in the Smith. A Rube-Goldberg fantasy of cams and levers that is the single most difficult trigger to regulate that I've ever encountered.
Perhaps it's poetic justice that the Fox and Smith used the triggers that they did. The Fox is not without flaw however. The opening of the barrels is checked by the cocking portion of the hammers stopping against
the inside of the action flat (I loathe the idiotic term "water table"). When the barrels are allowed to simply fall open, this puts great stress on the frame at that point and often results in cracking of the frame along the
barrel hook opening. Obviously this would be considered abuse but it does happen.