One way to rule out the fuel pump is to run the outlet line to a bucket and see if when you try to start the generator if it pumps gasoline into the bucket.
This is a somewhat dangerous test as you will be cranking a motor with open gasoline nearby so have a fire extinguisher and the garden hose on and close at hand.
But you should know within a couple of seconds whether or not your pump is pumping gas.
I don't know what you mean about the "fuel filter ticking away". The fuel filter should not make any noise. Perhaps you meant you can hear the pump ticking away?
Checking the filter is easy - remove it and blow through it. If you can blow through it, it's fine. Be careful and don't suck gas down your lungs, obviously.
Check the oil. These generators have an oil pressure sending unit and will shut down the generator if they don't detect oil pressure. You can disconnect or short out (I don't remember which) the sending unit to figure out whether it is the sending unit or not. You'll have to google on that.
These generators also have internal electronics that will shut down the generator or not allow it to run if it is not operating within certain parameters. Newer generators have a diagnostic display but not ones as old as yours (and mine).
You can try choking it to get it to run by putting your thumb partially over the air intake. If you can get it to run that way you've got a carb blockage.