Both suggested adapters described on (https://www.mikesarcade.com/cgi-bin/spies.cgi?action=url&type=info&page=9316.txt) have you either bypassing pin 21, or grounding it to pin 12. They both also connect pin 18 to pin 24 (Vcc). My initial thought would be to just switch pins 18 and 21. Why not just to that?

Is it because on 2716 pin 21 is Vpp (which is 5V for read/standby, and 25V for programming), and you don't want to risk having the programmer send 25V into pin 18 on the 9316? Doesn't the programmer only send the 25V when it is actually programming? When simply reading (or verifying) the chip doesn't the programmer simply send out 5V? So switching pins 18 and 21 would be fine under the read only mode? Is disconnecting or grounding pin 21 simply to be safe?