When I had an '06 3.0 TDI, there was no tank in the pump. The lift pump was built into the HPDP.
You only need compression and fuel to start a diesel. Can you undo a union on one of the injectors and crank to see if diesel leaks out. Should be a noticeable trickle from each?
Can you try a starting fluid such as Bradex Easy Start to see if it fires up for a second?
When cranking, can you feel pulses of gas from the exhaust?
What about the throttle that closes when you turn of the engine in the air intake, roughly behind the left headlight, made of aluminium - can you visually inspect the flap?
When I had a DPF blocked in 2010, a specialist removed it from the ECU but hollowed out the pre-cat only by mistake, so the DPF became totally blocked but would still start easily, just no power once exhaust pressure built up.