I've replaced about 4 starter relays on my 70 Challenger. Some new ones may be worthless because when I buy the replacement it works for a little bit, then it goes out again (even using different suppliers). I have used either a screw driver to make contact on both screws to start it or a remote starter cable, and then I can use the key thereafter, until it stops working again.

Should I be looking at another possible problem? All wiring seems to be intact, ignition switch seems to work fine, etc.

Any leads are appreciated.

Yes your neutral safety switch is giving you problems I bet. Mostly like needs adjustment by the shifter linkage.

Ground it at the relay to eliminate it for testing.

Next time it won't turn over try holding the key in the start position and move the shifter slowly from park to neutral. If it turns over then it is a NSS problem. Either a bad switch or shift linkage out of adjustment. Linkage doesn't need to be off much to keep the NSS from making contact correctly but it will still go into gear right.

and keep your foot on the brake while doing that

I appreciate the feedback. This sounds like the most possible problem. I'll fix.
thanks,