What could be the reason for failure in using a single-phase full-bridge square wave inverter to drive a coil for wireless charging?
The RD15HVF1 MOSFET is totally unsuited to the application and may indeed have been damaged by overheating. That device is specifically designed for VHF/UHF High power amplifiers applications.

It is not a good "switcher" and has an on-resistance of about 0.5 ohms on a good day. Driving it with Arduino outputs is not recommended either because of the gate capacitance takes quite large peaks of current every time you "switch" - this may damage the Arduino.

But it's even worse than this - the top three MOSFETs are also driven from Arduino outputs but are configured as source followers and this means the maximum positive voltage that can be driven from the source is pretty poor - maybe a couple of volts maximum. You NEED a proper driver circuit to avoid damaging your arduino and / or the MOSFETs (on load).

I'll also echo what was said about needing a capacitor from top drains to bottom sources to catch back-emfs when MOSFETs switch off.