How does this tesla coil design work?

At first sight this also can be an oscillator which outputs to the resonant circuit. In that sense it's not different than Slayer Exciter.

I do not say, that it cannot work, but I have some doubts. If it does not start to oscillate (for ex. wrong coil directions) the mosfet can easily get hot and destroyed due high Id and the Vds =20...40V DC. At least some current limiting should exist to prevent it

If it starts to oscillate, the gate insulation can get overvoltage. If this can induce the needed few volts peak-to.peak to the gate, there's nothing preventing it to be few times more, which can be too much.

This circuit shouldn't be impossible to simulate. Coarse equations for the inductances should be easy to find. The capacitance in the secondary coil is tricky to estimate. It's easiest to build one and add the parts that prevent the overheating and blowing the gate with overvoltage. That is to use current limited power supply and add a couple of zener diodes between the gate and source of the mosfet.