Sounds like the switch has failed. Using a LED should not matter - the circuit is dead simple. 12V power is at the oil light whenever the ignition is on. The light grounds through the NC oil switch, with pressure the switch opens and the light goes out.
Try taking the wire off the switch (on the oil pump, under left cover as you thought). If the light stays on, wiring trouble. If it goes out, switch trouble. Possibly the switch is "leaky", allowing a few mA of current to pass when it should be fully open. That wouldn't light the original filament lamp but would light a LED pretty brightly.