No-oil
and
Low-oil
are two different condition.
No-oil condition has nothing to with the level of the reservoir.(technically....obviously if there is NO oil in the reservoir then you'd expect the no-oil warning to sound , but that would be LONG after the low-oil warning horn would sound)
The no-oil condition is derived from the electronic section of the VRO.....that piece needs the Tachometer pulse to work, without it, it will never sound the horn (unless the electronics is faulty which rarely happens)
the electronics counts the tach pulses, and if after a certain number it doesn't get a reset, then it sounds the horn.
THE RESET: that comes from a little contact that is actuated from a tiny piston which is moved from the pressure of the oil being pushed through a tiny orifice. for every click of the VRO a shot of oil is injected into the fuel and this tiny orifice passageway, resetting the counter.
I know nothing of the System Check system and their Tachs, but it is NOT uncommon for them to simply poop out and give you wonky symptoms.
The horn too, can go wonky.
Does the horn chirp when you turn the key to ON? if so good (I think all the lights light up one by one as a system check)
Does the horn sound if you have the key to ON and then ground either temp switches' connector? if so good.
To be safe I'd do the following.
Disconnect the oil line from the VRO
remove the reservoir.
Dump the oil from the reservoir into another container. Inspect the reservoir's pickup sock for any debris.
With one end of the oil line into a container...pump air or alcohol through the line to be sure the line is clear.
Refill the reservoir, reconnect oil line at reservoir, other end into a container....pump the reservoir's bulb until you get a nice stream of oil into container....quickly reconnect oil line to VRO.
Try again and see if that resolved the issue (suspect it won't)
Then I'd look at the manual's No-oil Test Procedure and the Oil-consumption Test Procedure to be sure the VRO is sucking oil (which I suspect it is), and that the electronics will set off the no-oil warning horn.
We have some guys here that know the System Check tachs really well and can further diagnose whether it is simply a faulty tach.