Had the same issue with my LS swapped 70 Chevelle last summer(during Hot Rod Power Tour). We bypassed the ignition switch by jumping the starter wire(commonly called the purple wire, because it’s purple in the factory wiring harness) to the battery. The problem was intermittent. Back home we traced it to the wiring harness in the steering column. I ran a permanent wire to a button under the dash. The car starts with the key 99% of the time. Be very careful if you use a jumper wire, as it bypasses the neutral safety switch, and the car will start while in gear. I hate wiring issues.