In addition to wirefeed and gas, the third component in MIG welding is the arc itself. Does this remain on too? i.e., the MIG wire is always live too?
When the torch trigger switch is pressed a low voltage ( 12v or 24v DC ) is fed to a small relay on the control printed circuit board. This relay then applies power to the operating coil of a large contactor, which closes to send AC mains to the primary of the main welding transformer. This transformer supplies the arc, and also the wirefeed motor, via a speed controller on the PCB. The gas valve may get mains from the contactor, or low voltage from the transformer.
For all these functions to be enabled continuously, there may be a fault in the logic on the PCB, or the small relay contacts may be stuck or welded shut. The contactor could jam shut, perhaps due to magnetic grinding dust.
If you power the machine up with the covers off you may be able to hear or feel the small relay ( probably a white or blue plastic block on the PCB ) or the large contactor closing. Take care, the teminals and wires will be live with AC mains !
The contactor will have a plastic manual operating button in the centre of the top. If the contactor is not jammed shut, you should see this button move when it receives power - and this should only be happening when the torch trigger is pressed.