1. It sounds like you have an electric solenoid starter. Sounds like you are using a push button switch and an original ON/OFF switch.
2. You have it wired wrong. The ignition switch does not supply power to the push button. You need wiring advice so that your motor shuts off.
3. You are obsessed with lock cylinders and you are wasting a lot of time and money on a silly idea that you are getting 5 bad cylinders.
4. Get THIS
] switch, same as Jon_G pix, and use one of your lock cylinders in it and Mount this switch in the dash hole and also get rid of the push button. This switch has a far right momentary position for start.
5. I will advise how to wire once you settle down with lock cylinders. You have never actually explained what is broken about the lock cylinders. What is actually wrong with them mechanically other than "worn out".
6. Here in the Southwest we would make a concho belt from all those cylinders.