I don't know how he determined that the controller is the fault(I haven't gotten a response from him) but I'm thinking it's the display and not controller since the display doesn't light up or anything after pushing the power button.
The battery voltage at battery-to-controller is 40.5v.
The Scooter doesn't turn on at all at the display so it's still 40.5v after pushing the power on button at the display.
The battery completely disconnected still reads 40.5v
40.5v is low for a 36v battery, which should be 42v when full. Most likely it has some cells that are not fully charged (unbalanced), because the ones that did already fully charge are damaged, or old and worn out (so it takes them less time to charge than the rest, and less time to get empty during use). Sometimes leaving the battery on the charger for hours, days, or weeks (depends on how bad the problem is) can let everything become the same voltage so you get the proper full charge voltage.
if the controller and display are dead, is it possible that they can be diagnosed and repaired or is it just to buy a new set?
Depends on what's wrong. Not knowing what happened before they failed means some other basic testing needed first, to see how dead they really are.
Are there any fuses or switches between controller and battery?
If you get 40.5v at the battery wires on the controller itself, even at the moment you press the display power button, then that means the controller is getting power from the battery.
Can you take a good picture of the connection between display and controller? Then measure the voltages on each wire there, with your meter's black wire on battery negative, and put those on that picture for each wire, and post it here. If you also post a picture of the display (and the entire scooter) we can try to find out what model it is and what voltages *should* be there, and then see if it's one of those controllers that could be run without the display so you can verify if it works at all before getting a new display. (or trying to fix the one you have).
FWIW, the battery label *probably* indicates it's a 36v 5Ah, probably made back in 2017, which is pretty old for a cheap battery. The controller says it's from 2018, and is a 36v 19A controller with a 29V LVC, and is for 120 degree hall sensor spacing (so it probably isn't for a sensorless motor, unless it's what they call "dual-mode" where it autoswitches to sensorless if the sensors fail).