Please save the remaining three hairs on my head, car won't start

bbusby64

2001-Honda-Oddessy-215000mi-automatic transmission

I have been tearing my hair out for months and cannot get to the bottom of this, so far two professional mechanics have looked at it and are largely dumbfounded. I’ve poured out almost $500 for parts alone and am out of money patience and maybe even my mind. So far I’ve changed, the ignition switch, the security sensor, the main relay various fuses, the spark plugs, the starter, the battery, the cables, the fuel pump, the fuel filter, the alternator, and battery terminal connectors. when I turn the key, the fuel pump runs continuously at an acceptable pressure and the starter turns on, then stops, then comes back on, etc with no discernable pattern. Although as the battery drains the period of time that the starter comes on decreases while the period of time between pulses increases. When 12v is supplied directly to the starter it starts up immediately. If I jump the car at 13.8 volts I can usually get it started on one or two tries but its Russian roulette if I’m by myself with a start rate of maybe 30%. I’ve noticed that it’s also easier to start when the engine is cold. Please for the love of God someone help me. This car has been a nightmare and if someone is able to help me keep this thing running for 6 more months (until I can buy a new car) I swear I will never buy a honda again.