OK! Now here's the thing. 
It's fixed! Let me explain.
Right at the start I tried another two DVD drives. Neither of them worked. Or at least, so I thought/remembered. Taking another laptop apart and pulling out a compatible drive, I have just tried that one. It works fine - the HDD boots up and the OS reads the drive. I retried all the other drives and found that the original one and one of the other spares caused the problem, but the third one of the original ones I tried also works fine. ...
OK! Now here's the thing. 
It's fixed! Let me explain.
Right at the start I tried another two DVD drives. Neither of them worked. Or at least, so I thought/remembered. Taking another laptop apart and pulling out a compatible drive, I have just tried that one. It works fine - the HDD boots up and the OS reads the drive. I retried all the other drives and found that the original one and one of the other spares caused the problem, but the third one of the original ones I tried also works fine. ...

It's fixed! Let me explain.
Right at the start I tried another two DVD drives. Neither of them worked. Or at least, so I thought/remembered. Taking another laptop apart and pulling out a compatible drive, I have just tried that one. It works fine - the HDD boots up and the OS reads the drive. I retried all the other drives and found that the original one and one of the other spares caused the problem, but the third one of the original ones I tried also works fine. ...