Addressing your points in turn:
1) Handle ties do not make a BR breaker pair common trip (2-pole BR breakers have an internal common trip mechanism) -- see footnote 2 on page 84 of Eaton's catalog.
2) Common trips are generally used in 240VAC circuits -- while what the previous installer did with the pump circuit is not wrong as per the 2014 NEC 240.15(B)(2):
Grounded Single-Phase Alternating-Current Circuits. In grounded systems,
individual single-pole circuit breakers rated 120/240 volts ac, with
identified handle ties, shall be permitted as the protection for each
ungrounded conductor for line-to-line connected loads for single-phase
circuits.
common trips are generally used on all 240VAC circuits (it's hard to get a non-common-trip 240VAC breaker in a typical form factor), and are needed in mixed 120/240VAC circuits for the reason you observed, as per NEC 240.15(B):
Circuit Breaker as Overcurrent Device. Circuit breakers shall open all
ungrounded conductors of the circuit both manually and automatically unless
otherwise permitted in 240.15(B)(1), (8)(2), (B)(3), and (B)(4).
(The exemption in (B)(1) is for multiwire branch circuits serving only 120VAC loads, while (B)(3) and (B)(4) are irrelevant to residential practice.)
3) You are correct -- this is documented pictorially in the table headings on page 82 of the Eaton catalog, so it's easy to miss.
Get a different electrician and/or make your power company come back. It sounds like your losing one leg of your service.
The reason your cooktop control sends it into remission now is that it starts by backfeeding the bad side of your service which heats up the connector that is burning. The heat (or arc) acts as a temporary weld that makes it work again. For awhile.
The next thing to happen will be that the heat control on your cooktop just acts like a really big dimmer switch for all the circuits that go off when your issue occurs.
Yes, fire is a possibility but it will likely be contained to your meter can or breaker box (possibly a pole or underground connection).
Best Answer
Sadly, there just isn't much data to go on
Unfortunately, very little is known about the behavior of Wadsworth breakers; the original manufacturer is nothing more than a brand name that's been tossed about the industry a few times since they stopped being a going manufacturing concern, and while there are replacements out there, there is basically no manufacturer data about their performance, either. Nor are there any independent test results; unlike the FPE fiasco, which motivated a significant independent study effort (the work of Aronstein et al) that also swept in Zinscos and their checkered history as well as several more mainstream brands of breakers to serve as a reference point, Wadsworth breakers have not been analyzed in-depth by anyone outside of UL (back then) or ETL (more recently).
You can't really test a (small) breaker in-situ either
Atop this dearth of information, there is basically no way to safely test a miniature (loadcenter) breaker while it is installed. This is because most breakers consist of two parts: a thermal trip that protects wires against overheating caused by overloading of circuits, and a magnetic trip designed to catch short circuits and other such catastrophic, high-energy faults.
With the thermal trip, the primary hazard of testing it while the breaker is installed is that the circuit itself will simply fail open at a weak connection under an overload test current (say, 200% of breaker rating), creating a hard-to-diagnose fault. If you carried on long enough without the breaker tripping, the wires would eventually overheat and create a fire hazard, as well, which could be a problem with relative unknowns like your Wadsworth breakers. (It's less of a problem with modern wiring like the THHN used for your homeruns than it is for old cloth wiring as modern insulation withstands heat better and thus as better safety margins against overloading, but it's still possible to do.)
However, the primary problem is that the thermal trip is rarely needed to safeguard a circuit from damage; instead, under high-energy fault conditions, such as those created during a short, it's the magnetic trip that needs to operate in order to instantly disconnect the circuit and thus stop further damage cold in its tracks. These levels of fault current (at least 10 times the handle rating) are much more damaging to the circuit as a whole than a simmering 200% overload is, and there is very little limit on how high a breaker's magnetic trip can be set.
As a result, there isn't any good way to test a loadcenter/miniature-type breaker in-situ without risking severe circuit damage, and the high-powered equipment required to test a breaker's magnetic trip function can't realistically be brought out to a worksite either, even if pulling the breakers in question out and sticking them in a test machine was in the cards.
There really isn't a good path forward
The other part of the problem here is that it appears that we aren't looking at a simple breaker panel here, but some sort of multi-way meter-pack that contains a set of electricity (watthour) meters and associated small breaker panels. As a result of that, and the space constraints associated with that, there's really no way to fix this without essentially rewiring significant chunks of the building.