It sounds like the fuel system is slow to build pressure and/or is not holding pressure. In a minute or so of cranking the system might be building just enough pressure to run the engine, no more. If you run the engine only for 15 sec, maybe not enough fuel flows in this time to fill the whole system and provide much overpressure (ahead of the fuel pressure regulator). Then by the next time you start up the little fuel that was accumulated during the short run would have leaked down (backflow through the pump, injector drips, leaking regulator, etc) - even a little or normal leak might allow too much fuel to be lost. On the other side, if you drive for a while, there would be more accumulated fuel in the system, so enough for the next start would stay in it longer.

So it might be useful to measure or have measured the fuel pressure after running the engine briefly then stopping it - if not holding it would point to the regulator or pump or a leak. I would not think a partial block of fuel flow would let the car run normally during driving.

It's a problem for this explanation that you can get the car to start with the pedal floored - if the injectors are cut off the car should not start regardless whether the fuel pressure and flow are normal or not.

In some older Volvo models there was a fuel accumulator or a 2-stage pump arrangement and you could get a similar symptom to yours if one of these failed.