Ford Repair: 2001 windstar starting, fuel pump relay, electrical mechanic

Question
High mileage (180K). About 2 weeks ago my wife said the van would not start shortly after she returned home. I hopped in it, turned the key and it started right away. I tried repeatedly and it started every time.
Then a few days ago it did it again. This past week end it became frequent.
I can now predict when it is going to start.
When I turn the key on, if I hear a humming sound (I think fuel pump) it will start. If I turn the key and hear nothing it will not start. Repeated key OFFs  for a few moments , then ON, then OFF for a few minutes eventually results in a 'hum' and a start.
I thought I had a bad fuel pump relay so I switched the horn relay with the pump relay. No change. The horn works and the van starts predictably as above.
A busy garage was able to attach OBD tool and found no error codes.
A friend helped my wife last week. He said he cranked and held the accelerator pedal part way down and off it went. ( I can not duplicate this. I do not feel it does anything.
Fuel tank is full. No check engine light on. ABS  light and brake light almost always on (known issue).
Is there another fuel pump relay?
I am thinking - starter with excessive draw and not enough voltage for ECM? key switch with ON circuit not functioning when in START position? some type of fuel delivery problem? failing ECM or other e device, failing fuel pump.
I wrote to Tony Florence last week but he was unable to reply and is now out on vacation.


Answer
I think you are right, it is an electrical problem. Holding a late model car's throttle down, or depressing it at all actually, is highly un-recommended. Ignition switches are usually not hit and miss like that, but don't rule it out entirely. What you have to do is this, if you want to continue trying to self diagnose. First thing is patience, patience, and more patience. You are going well beyond shade tree skills into electrical mechanic diagnostics. If it was my families transportation, quite honestly, I would drop it off at a Automotive Electrician. This is not your normal mechanic, these guys are wicked mart, and always have a full lot. If you want to troubleshoot yourself then start here. When you car fails to start, test the voltage at the fuel pump. Work your way forward from there. We assume that your fuel pump is not cutting on, but we don't know that for sure until tested. You can rule out the relay, as you already know. The only other option is the ECM as you stated. However, you should rule this out last, if at all possible.