I have a 2000 Jackaroo (Trooper) with a 4jx1 motor.
These motors use a hydraulically powered electrically actuated injector system. Engine oil is used for this purpose.
The choice of oil is very critical to the operation of the motor and it has been found in the cooler climates that a 5W-30 oil is required.
The reason for the slower starting of the motor- like the six seconds you quote is due to need for the injector supply pump to build up pressure to adequately operate the injectors. The second filter on the LHS of the engine is for this pump.
The main reasons I have had or have heard for the engines not to start are
1 Dead or dying batteries (pretty obvious)
2 Leaky injector o'rings allowing gas or air into the fuel system. This is usually picked up by placing a piece of clear plastic hose into the fuel return line and looking for air bubbles. Most countries conducted a recall to fix this problem. In extreme cases I have heard that the oil can run into the cylinder through faulty seals/seats and cause engine run away and subsequent destruction.
3 Leaking mechanical fuel primer pump. I think there is an excellent description of this in the archives of this site.
But same deal, will allow air into fuel system and allow the fuel to flow back to the tank when engine is switched off.
4 loose or broken oil pick up/strainer for injector oil pump.
This happened to mine with about 38,000 km on the clock and took the local dealership a month to find it and repair. In my case it was only loose but I have heard of them breaking off. The sump pan needs to be removed to check for this.
I hope that these ideas may help your mechanics solve your problems

Regards Craig
In regards to the fuse thing ....If you blow the brake light fuse the car won't start either. I know a person who had their truck towed for 1000km to the nearest dealer to find out that he had just blown a brake fuse.
There is a fault light but you need to find which pins to short on the harness connector which is located somewhere in the dash behind the radio or ash tray
The number of times the malfunction indicator light flashes gives the identification of the fault. A search on the internet may help you find the codes.