The only thing I can think of is the starter or battery. You say the starter is turning the engine but, doesn't sound like it is. So, it is not turning it fast enough?

If it is turning don't worry about flex plate, that is what the gears of the starter engage.

2006 2.2L does not have VVT.

Checking the marking on the chain is not as easy as it sounds. They only line up 1 time in something in the neighborhood of 64 (or more) revolutions.

The codes are VERY important.

Check for spark.

 

You stated you had the valve cover off and the chain was "good". I would suggest taking the valve cover back off, try starting the engine and see if the chain is moving at the cam gears.
A possible broken chain (broken at the lower end) may not be out of question. That may also account for the odd starter sound you're hearing.

Just guessing here but that would be my next move if it were my car.

Fwiw. If the chain has broken, you'll have bent valves or worse because the Ecotec 2.2 is an interference engine.

 

I have the valve cover off and started the engine. Both chains are rotating. Is the cam gear towards the back of the engine the exhaust and the front one the intake? I keep turning the engine until the marks line up? And should I put back the old crankshaft position sensor?

 

Is the chain nice and tight? If you're going to check the timing, you need to be able to see the mark on the crankshaft also. All 3 marks have to line up. The exhaust cam is in the rear, should be at 10 o'clock. The intake cam, in the front, at 2 o'clock. The mark on the crank should line up at about 4:30 - 5 o'clock position. The special colored mark will be on the intake cam.

 

Like this, the crank key slot at 12:00 o'clock #1 at top dead centre.

 

check fuses, OBD2 for any codes, battery , flywheel, starter solenoid, fuel pump, fuel filter etc. Im thinking its probably your fuel pump. When that goes bad, the car will often crank but will not start. OR, it will start but die a few seconds later.

 

It's a month old thread and the OP hasn't been back. Besides, all the general things you suggest to check have all ready been discussed and checked by the OP.