I have not heard a variator cause these issues.
If the car idles ok, but stalls when you try to rev it, I would suspect an air leak in the intake rubber hose. When the engine accelerates, it moves a bit and bends the intake hose, causing any cracks to open up and suck air. whereas if the crack is small, it won't cause a leak during idle or steady revs.
Another problem might be the battery. If it's totally dead and won't take charge it might affect the injection. I understand that you were able to start the car with the battery after starting, so probably not the battery, but you might want to run the car with all other electrical devices turned off - lights, fan, radio, seat heaters etc. and see if it makes any difference.
Any problem with any of the sensors would most likely light up the injection light, so I would look for the trouble elsewhere if the injection light works alright (during startup) but is not lit.
One more thing - where did you store the car? Any chance the mice might have chewed the wiring?
EDIT: A new rubber hose may be difficult to find, but if you wash it first it can be easily fixed with Sikaflex or similar rubber glue.