67 barracuda starting issues

Sounds like oily/contaminated/compromised connection, either outside that neutral safety switch, or inside it.
Even when those connections are good that ground still has to have a path all the way back to the battery negative post. There is ample metal to metal contact though. Most of the time everything chassis grounded does work even where the chassis ground strap between engine and firewall is left disconnected after engine R&R. For now lets assume the fault is right there at the neutral safety switch. I'll mention one case where that wire was cut and pieced with household wire nut one place and electric tape at another. Was a chevy truck but same intermittent failure. Just never know what we'll find in these old used vehicles.
If yours has OEM trans, etc.., it's a somewhat less common, more difficult to find, single post/single function switch.
A good tech' would recognize what his tester is showing, buy you a switch and install it. What you currently have is a unsafe patch. Be sure the shifter is in park before turning the key until its properly repaired.