I had similar vibration and it was the Fan Clutch going out.
When engine is cold start it and test for vibration, when you know where it vibrates, RPM wise, shut off engine and remove(disengage serpentine belt), restart engine and test for vibration, battery light will come on maybe even CEL, but it won't hurt anything to run engine without water pump for a minute.
If vibration is gone then my guess would be fan clutch.
If it only vibrates when warm then I would bet money on the Fan Clutch, but your money not mine, lol.
Back to the slow start, you have sequential injectors, all will have power(12v) when the key is on, but none will have a ground.
Computer grounds each injector to open it.
If one of these six Ground wires had a short to ground then injector would open on it's own.
You could use a test light or meter with key on and test if any injectors have a ground.
On a cold engine it would start but not quite as fast, cold engines like Rich mixes.
Warm engine doesn't like rich mix, would take some time to start.
Pressure test should show this, pressure would have a steady drop with engine off key on as fuel leaked out via the open injector.
Have you tried the Key on/Key off test to prime the fuel system?
Turn key on count to 3
Turn key off
Repeat 3 times, then try to start, this cause fuel pump to prime system 3 times.
If it starts right up then I would thing fuel pump issue, might not be the pump but a leak in the delivery system in the tank.