Car stutters on takeoff

There are several posibilites;

The intake and throttle body could be dirty, as well as the Idle Air Control Valve, (IACV), and the Fast Idle Thermo, (FIT). If the Intake tract and throttle body are dirty, that can affect performance. If they are dirty, the IACV, (which controls the idle as the engine load increases from; power steering, air conditioner, headlights, other accessories, etc. The FIT controls the idle in sort of the same way, but it works when the motor is cold like a choke and it can fail which can affect the idle even after the motor is warmed up.

There are also emissions stuff like Exhaust Gas Return (EGR), Positive Crankcase Valve, (PCV), etc. Which can have some affect but not as much as the other items.

Howver since the FIT failed, it may have failed due to the Exhaust Gas Return, EGR. Disonnect the vacuum line to it and see if it stalls. If it stalls the EGR is dirty and needs cleaning or replacemnt.
Check for vacuum leaks and have the fuel pressure checked.

The pip into the EGR should lead to the exhaust somewhere. It could be the Fast Idle Thermo Valve you are looking at which has a coolant hose attached to it.