At a given (constant, sine wave, AC) frequency, there are two parameters to every (two-terminal) component: resistance and reactance.
An ideal inductance or capacitance has no resistance. Real components always have some of both.
What's more, the amount of each changes with frequency.
A component we call a "capacitor", is one which behaves like a capacitance over a useful frequency range. And that frequency range might be very different for different kinds! A small ceramic capacitor works very well from ~kHz to 1GHz or more; a supercapacitor might only be usable at very low frequencies, ~Hz to uHz!
So first of all, be careful about using components, because they are only useful (for stated purpose) over a range.
(Your MKP capacitors are probably just fine, here. If they get awfully hot -- too much resistance -- try using pulse-rated or snubber type capacitors instead.)
Inductors are interesting, because they couple to other inductors nearby. And if those inductors happen to have resistance connected to them, then the first inductor also looks like it has more resistance. This is simply transformer action: a load on the secondary, is reflected as load on the primary.
So, what's happening is, the resistance of the workpiece is coupled to the work coil, and since the transistors are driving it alternately, they alternately supply power into that resistance. The current through the coil is alternating in polarity, but because the transistors also alternate, you get positive and positive current draw from the DC power source. Which means we're drawing real power, and dissipating real power in a resistive load.
It's a nice property, of this type of circuit, that it has a constant-voltage characteristic. That is, the load on the power supply varies with the load on the coil. It doesn't draw much power when the coil is unloaded, it just sits there oscillating. When you add a load, almost all the extra power that's drawn from the supply, goes right into the load! (Not all, because of losses in the transistors, coil and capacitors.)
Tim