questions on push button ignition/

You have yellow and brown both hooked to the output of the push button? Can't do that

So far as why it won't crank, go out under the hood, and rig your test light with clip leads.

Find the yellow on the start relay, You have 1 or 2 push on terminals on your start relay?

If you have 2, the neutral safety switch is probably not getting ground. Check your yellow and see if you have power with the push button down.

So far as the brown to the bypass, that has to be isolated That's why I suggested you use the later model Jeep starter relay. You can hook the brown bypass circuit from the indicated terminal on the Jeep relay go the coil +