The LM1881 is a sync separator, its function is to extract sync from a composite signal and I don’t believe it can generate sync on its own.
In the Cadet I the extracted sync components are used by the ATMEGA to derive vertical sync and odd/even clock signals. The horizontal sync then is extracted from the composite sync signal using a 74HC4538 multivibrator IC, and has its phase compared to the hsync signal generated by the ATMEGA. The phase difference goes through a passive lowpass filter, then is used to modulate the frequency of a crystal oscillator, which acts at the master clock for the ATMEGA. This creates a feedback system that’s a type of phase-locked loop which ensures that the ATMEGA’s horizontal sync tracks exactly the frequency and phase of the incoming sync signal. This way the ATMEGA generates the horizontal sync, vertical sync, etc. at all times. This design is described in more detail in the “Circuit design notes” section of the manual, right after the schematics.
The only output from the Cadet I that’s not generated by the ATMEGA is the RCA output, because this is just a buffered copy of the input signal. This is useful for taking a camera signal and deriving sync from it while passing the unaltered signal off to some other part of your rig for processing.
Sorry to hear this has been giving you so much trouble! As others have said if you’re able to program the ATMEGA at all, and you’re otherwise seeing good signals on the scope, you’re probably almost there. Here is the command line that worked for another builder – it looks like your command line is missing the fuse settings. Lars posted a screenshot of the required fuse settings in the post right above that. I’ll submit a pull request to the LZX firmware Github with this command line, hopefully that makes it easier to find.