Based on the video, the concern is a no crank no start, correct? The first time you turn the ignition to start I hear a noise, but it does not sound like the engine is cranking. If that is the case, then the dealer improperly diagosed the issue. A purge valve or EVAP (carbon) canister will never cause a no crank condition.
As others have mentioned, it could very possibly be the starter, especially with headers. Other possibilities include the ignition switch, the circuit to the starter solenoid ("S" Terminal), the Transmission Range (TR) sensor, the starter relay, or the main power (B+) and ground (starter case). It does not appear to be Body Control Module (BCM) or PATS (Passive Anti-Theft System) related because it does not display "Starting System Fault" in the Instrument Cluster.
Start by inspecting all of the connections at the battery and starter. They should be clean, tight, and free of any corrosion. If there are no connection issues, wiggle the shifter or move it to Neutral and try to start it again. If it starts on the first try, try the same test on the next no crank and see if it is consistent. If moving the shifter gets it to start, you may have an issue with your TR sensor.
Do you have access to ForScan? The PIDs are very helpful when diagnosing issues such as this.