Well, your modification would very likely worsen the safety.
Either you replace only the plug, it would be just that, a new plug, with the GND not connected anywhere. That makes it worse by giving a false belief that the appliance is grounded.
Or you decide to replace the cord as well. It would mean opening the chassis, finding a connection point somewhere and soldering or connecting the GND wire there. However, you first, can't be sure that the connection point is "good enough". Second, opening the chassis may damage some insulation. Third, the device, as is, was tested against various failures and use conditions, your modification would not be.