Not having fast enough driver at that frequency
That still sounds high. Typical gate resistor values are on the order of 5-15 ohms.
I agree that the gate drive resistors should be a much lower value.
For a switching application there should be more modern MOSFETs than the rather old IRFP250.
Sorry gentlemen but this is NOT a switching application, it's a linear circuit!
(Indeed, one can attempt to switch-ify the circuit, but it quickly becomes a cluster, until you give up and embrace the full digital equivalent, a resonance-tracking PLL. At that point, you might as well drop the push-pull configuration as well, and use a more traditional bridge circuit.)
FWIW, using newer transistors may well have the downside that they can oscillate at 100-400MHz. Usually, a ferrite bead on the gate and source is enough to deal with this, but that's typical of a low impedance drive situation, and YMMV here. If you have a 100MHz scope, you might miss such behavior! But with a scope much slower still, you'll definitely be able to verify if the IRFP250 is getting adequate gate drive (i.e., it's not slew rate limited).
Note also that the circuit requires shorting-mode commutation. It's a current-source inverter topology, hence the series inductance in the supply. Ideally, the high impedance supply should extend all the way to DC, using a constant current bench supply, say -- unfortunately, almost no bench supply has a true CC output, they inevitably screw it up with a big stupid capacitor across the output terminals, for no good reason.
Many of the quirky properties of the circuit, only seem quirky because we are not used to seeing the current-mode transformation of traditionally voltage-mode circuits. Hence, instead of dead time between switching, there must be shorting time; instead of CV supply, CC supply; bypass inductor instead of bypass capacitor; push-pull instead of half-bridge; etc. (If you're curious, it's a fun exercise to enumerate the series-parallel, voltage-current transformations of conventional switching topologies. Some are in use under a different name; many are not!)
IRFP250 is a pretty damn solid MOSFET, so if it's cooled adequately, there must be something else going on. Current surges? Perhaps due to the above reason (excessive current available from the supply)? Is the oscillation being quenched (low Q load) resulting in shorting out the supply through the FETs? Is the gate drive accidentally being turned off before the supply inductance has discharged to zero current? (Again, shorting mode operation: the gate drive must be available before and after drain power is applied!)
As for counterfeits, if you bought them off eBay or Ali Express or whatever, you've no one to blame but yourself. Suspect 100% of cheap crap bought there. If you don't have the time and money to test and verify suspect components, buy from a reputable distributor.
Tim