A question worthy of much discussion I feel. I would say - my opinion here- that the purpose of the breadboard end primarily is to cover the end grain and dampen moisture exchange at the table end. The idea that a breadboard end, at least if it is the same thickness as the table top, will stiffen the top against movement (cupping) is probably wildly optimistic.
Also a note: the 'Ming Inspiration' table did NOT have breadboard ends as it was not constructed with a table top slab but with a unique type of frame and panel construction. The frame and panel in most respects is simulating the look of a slab top with mitered breadboard ends on that table. Brian's observation about how that table design accommodates movement is correct.
I built a table many years back which had a bubinga plank top about 30" wide with mitered breadboard ends. It seems to have worked okay over time, however it spends its time in an environment with a narrower seasonal swing in relative humidity than other places.
I think that the use of mitered breadboard ends is a more workable proposition given these conditions:
-placement in an area with narrower seasonal RH swing
-using a table top wood with a quartersawn orientation generally
-using a wood species for the top which doesn't move much seasonally
You need to be able to calculate the amount of movement to expect, given the species, moisture content, grain orientation, and RH range in the environment in which the piece will be placed. You need this information in hand at the design phase. The top will move, and there inevitably will be times of the year that the miter is slightly open or cranked up tight.
Of course the attachment of top to the breadboard ends must allow for the top to move relative to the ends. Assuming you follow the standard strategies in that regard (tenon pinned in the middle, and tenons toward the outside with elongated pin slots), one thing that will tend to happen at the mitered interface is that table shrinkage will tend to push the mitered end of the breadboard away, bending the breadboard ends. Of course a gap would open up at the miter, however this is preferable to a crack being propagated on the tabletop. Still, that crack remains a risk.
" I seem to recall several tenons with a "tongue" that continued up the miters and corresponding mortises/dado on the breadboard end. "
The purpose of the tongue/groove at the miters is to keep the upper and lower surfaces of the pieces more closely registered to one another. Without the tongue, there would be room for the mitered portion of either the table top or the breadboard end to warp or cup slightly, leaving a non-smooth interface between the parts along the miter.
I've thought about this issue a great deal. Initially I designed the coffee table (detailed in the 'Square Deal' series of blog posts) with mitered breadboard ends, then, realizing it wouldn't work so well, went to an arrangement of having a miter at only one end of the breadboard at each side (pinned at the miter and floating everywhere else it was connected), then moved to the idea of the hammerhead keys, which I feel is closer to an 'optimal' solution. The table will be in an environment with modest seasonal RH swing, but the top has a significant portion of tangential grain, which will means a possible 1/4" of movement in the wood used for the top (bubinga). I was apprehensive about the performance of mitered breadboard ends in this context and avoided it.
Really, the time-tested classic solution, if you want to control movement and have the parts well fitting at the miters, is to move to a mitered frame and floating panel arrangement. The trade off, of course, is a table surface with crud-catching grooves along the panel edge and a less seamless visual than you would get with a one-piece top. If the top is to be constructed from a glue up of boards however, then I'm not sure if it is any 'quieter' looking than the frame and panel with a one piece or two-piece (etc.) panel.
Balance that form of construction and its drawbacks/advantages off against the offsets that will happen at different times of the year with a breadboard end relative to the slab.
With the 'Square Deal' coffee table, I had the unique opportunity of using a single 38" wide bubinga slab, and it was a technical challenge for me to solve the 'breadboard' issue. When I experienced the top cupping at various junctures during the build, and was getting quite nervous about it at some stages, losing sleep and all, I started to wonder if it might not have been better to have gone to frame and panel. A panel can have thick dovetail battens fitted to the underside which will definitely keep it flat. I controlled that top and kept it flat with clamps and weights, but it was quite nerve-wracking at times, believe me.
Looking at Ming tables generally, a slab top seems to have been confined to use on tables which were relatively narrow - in fact, it was highly desired to have a thick slab top on such tables, though the weight of that top posed a challenge for 3-way mitered connections at the table corners. Few of those tables have survived - a fine example is at the Minneapolis Institute of Arts. These narrow tables often had everted flange ends, which were also mitered at the top's end sides, but I think the narrowness of the table tops in those pieces makes this construction a more viable proposition. Any wider Ming tables I have seen, like painting tables, are always made with frame and panel construction. It is possible that some were made with slabs, but none of those have survived.
Looking at tables in museums from the 1700~1900 period, a very common thing to see on any table with slab tops or slab folding leaves, etc., is warpage and cupping. It tends to suggest that it is not a method of construction which keeps strictly flat over time.
So, I guess I tend to feel that the frame and panel approach is the most ideal, while being clear on the drawbacks. You can make a slab top work in various way, but drawbacks also abound.