Maybe the car senses a Chevrolet freak has custody?
The horn relay has nothing to do with the starter circuit. It operates the horn, is all. You'll need one to pass safety inspection. They're on ebay all the time and work on Buick-Olds-Pontiac from 58-68. Trouble there is you're fighting the GTO crowd for them, esp for NOS Delco parts. If you're not terribly concerned with original appearance, you can adapt a Chevy or Ford relay to your wiring until you can find or afford the correct horn relay.
Have you checked to see that you have 12V at the ignition switch? Mid-60s Olds sometimes melt their connectors and cause all sorts of havoc, but you said it looks ok. You want constant 12V at the RED wire.
You say everything works in ACC. Try them in RUN. There are two 12V busses in the fuse block- one fed off battery (always hot) and one fed thru the ignition switch (hot in ACC or RUN). If it's hot in ACC but not RUN, ignition switch is suspect. You want 12V at the PURPLE wire terminal with switch in "START".
If you have 12V to the ignition switch, the problem has to lie in the ignition switch, neutral safety switch, starter solenoid, or the associated wiring. You're dealing with 40+ year old wiring that if you're lucky hasn't been boogered with by someone who had no idea what the hell they were doing. That's a big reason I tend to stay away from Chevrolets and musclecars- they
ALWAYS have boogered wiring. I call stuff like that "previous owner's stamp". Olds and Buick are less likely to suffer from it than Chevelles, GTOs and Mustangs, but I've fixed my share of corrupted wiring on all of 'em.
Here's something off the wall but it bears checking. Are the right wires on the right plugs at the neutral safety switch? EZ test- see if the backup lights work with shifter in R. These are the GREEN wires. If that checks out, jumper across the two PURPLE wires and see if starter operates. If yes, NSS is faulty or out of adjustment.
On the carb linkage switch- one side of it controls electric downshift on the Jetaway, and the other set of wires controls the switch-pitch torque converter. Has no effect on your problem, but needs to be adjusted correctly for transmission to work right.
The best investment you could make right now if you plan to keep the car is to buy the 1965 Olds Chassis and Body Manuals. They will tell you about anything you need to know to service your car.