Jonathan, while I've got the "momentum", I'll elaborate why I'm so suspicious about this coil.
All transformer coil endpoints are marked. When you apply 2 AC sources to 2 separate coils on same core, it becomes vital. AC power source depends on transformer to have some specific inductance, this causes impedance (analog to resistence in DC domain), and that defines how much current can go into coils. When you connect power to coils wrong (opposing) way, there happens so to say magnetic short circuit, coils cancel each others flux and inductance goes to nada, this results in impedance collapse and nearly unrestricted current flow between power inputs. Basically, it is very similar to short-circuiting the AC power source. This happens occasionally when careless, and can (literally) blow out equipment and cause fire.
And this patent coil reminds this mishap enormously.
Inductance alone has linear dependence of impedance to frequency, all the way from DC to infinite frequency. Linear. There is no resonance. Resonance occurs when there is phase shift large enough to cause delayed output to coincide with input. In circuits this requires capacitance, or quite long delaylines. For eg, if you have 200,000km wire, then it takes 1 second to reach the end. If your input AC is 1 Hz frequency, and its other end is brought back to your power source, you can get resonance. For shorter distance, frequency needed is higher, waay higher. Incidently, impedance goes up with frequency and inductance, and that requires ever higher input voltage to get same power transfer.
Tesla needed several things, very huge inductance, normal accessible input voltage levels, quite high frequency. This is quite conflicting set of requirements. So he came up with this coil. Its basically short circuit. imo. But currents inside aren't perfectly balanced. Transient magnetic fluxes have time differences. With sufficiently huge coil and inductance that can be brought down to manageable delays and thus frequencies. When that happens, we maybe could even talk about some resonance. But it depends heavily on material then and is nonlinear. Tesla's goal was to transfer energy over distance, and he basically used Earths capacitance for resonance. So his coil had resonance when interacting with environment.
This all is very different on the table and in small setup. After inductance collapse occurs, there is a short period of time when currents inside coils and iron core fluctuate wildly at random. That can cause heavy RF interference, while benefits are questionable. There are other winding methods to produce even better magnetic fields. For Tesla, it was a sort of energy pump. While inductance collapses, power source is detached from the coil's output. With capacitance behind it, the energy that got into the coil was transferred out at that combined resonant frequency.
When it comes to electromagnets, then shorting power source is a bad thing, wastes energy. RF intererence is illegal thing, and wastes energy too. High frequency currents is unwanted thing, core efficiency goes down. Inductance of coils is friend actually. It can be managed and brought to optimal levels differently. Thats why I think this coil isn't used widely.