Yes the series 60 can be hard to start when run out of fuel.
If by filling the the fuel filter on the engine and it will not fire,
It sounds like there is air in the head.
You don't want to put a in line electric fuel pump in this can damage the seals on the engine fuel pump.
On the back side of the head there is a return line.
You can loosen the line at the elbow fitting here
Wrap a rage around the loose fitting,
Turn the engine over for 15 seconds.
Check the fuel filter on the engine,
Keep the filter full.
Repeat turning over for 15 seconds.
After the 3rd or 4th try the rag on the return line should be wet,
You can now remove the rag and tighten the fitting.
If you want for a start assist use WD40 and spray onto the air cleaner.
This is flammable a lub and is safer then start fluids.
If your having fuel problems still and the filters are not filling
loosen the line on the fuel pump going from the pump to the filter on the engine.
See if there is any fuel coming from here?
You do have a fuel pump problem.
The fuel pump is on the back of the air compressor
Let me know if i can help more!