I'd check the wiring.

Definitely sound like a wiring issue. The connections on the starter motor would be my first suspect.

For the carb, post some wider pics and some close-ups of the linkage and the front bottm of the carb. You may not be able to adjust the idle mixture as some of this vintage are set at the factory and sealed. You should be able to adjust the idle speed though.

There's more than one millenia siting around local junkyards here in the Merrimac Valley.

Last time I was picking through the diesel Mercedes section of Jack's there was a very

complete millenia under a 929 in the Mazda stacks.

This is the same junkyard that crushed a '69 Camaro shell and semi-solid BMW 6.

Might want to replace the wiring to the starter anyway. And the alternator one too for the minimal cost. I had one car where I could not figure out why it would not start sometimes. Replaced the battery, alternator,wires LOOKED good. The operative word here is LOOKED good. There was corrosion inside one where you couldnt see it. Once that was replaced, no more problems.