Your symptoms indicate more than one possible issue. Slow to start could be a failed fuel accumulator... easy to test on the car and easy to change. Search for threads started by me for all the details including my destructive analysis with photos of the cracked diaphragm.

Mercedes advertised that if the "e" part of the KE system failed the engine would continue to operate. However, they failed to mention that all the cold start enhancements are in the e-part of the system, so if there's an issue on the e-side, cold starting can be very difficult.

The fuel pump should energize for about one second when you turn the ignition to the "On" position. Have a helper listen. Run the test multiple times to verify that the fuel pump relay is working consistently.

If everything is normal the engine should start and initially idle at about 1000-1200 in neutral depending on coolant temperature. You can't "kick it down" like an old carburetor with a mechanical auto chock since the cold idle speed is controlled by the e-part of the system based on engine coolant temperature, so check the coolant temperature sensors.

I don't think the cold start injector is energized below about 32F, so if you're cold starting above that it shouldn't be an issue.

As previously stated, if the OVP, which supplies power to both the KE and ABS controller is bad the ABS light will remain on after it starts up, so it's probably okay, and unless the system voltage is outside the range of 13.5-14.5 volts with the engine idling the regulator is likely okay.

The KE system is complicated and parts aren't cheap, so swapping out parts apriori can get expensive and time consuming... doing all the diagnostics you can to try to isolate the problem is a better way to go.

Duke