Hi Ellen! How exciting it's getting with your trip planning!
Here's my 2 cents on things.....
I think your decision to not rent a car is a good one. The really nice thing about the Rome to Amalfi Coast trip is that there is no NEED for a car. Public transportation is EXCELLENT so why not "When in Rome" and do like the Romans/Italians do and use the public transit? It's fun and easy (truly, REALLY easy, I NEVER use any kind of public transit in the USA, but had no trouble figuring out the trains/busses/boats).
Driving on the Amalfi Coast drive....meh....I think the danger of it is really overstated. It's a winding road. OK, whatever. It really didn't strike me as dangerous. BUT....as we were on the bus and my husband was relaxing and looking out the window he commented "I am SO happy to not be driving so I can actually SEE things!" and I (as the family designated navigator) was also glad to be able to sit back and enjoy the view from the public transit without having my nose buried in a map.
If your husband isn't 110% comfortable with a stick, he would be MISERABLE trying to drive when in southern Italy. Not just the winding roads, but having to back up in narrow confines to let a bus by, tight maneuvering in villages, everyone driving fast and confidently and CLOSE to each other, lots of going up hill and around corners......imagine your husband trying to stop the car and then put it back in gear when you're stuck behind a bus on a narrow mountain road and you have another car sitting impatiently 2 inches off his back bumper. Boy, that would be a total trip buzzkill and I guarantee you he would be sweating and stalling and rolling back into the car behind you.
But probably the SCARIEST thing about having your own car is TRYING TO FIND A PARKING SPOT! When you get to these towns you'll see what I mean. I cannot even imagine the heartburn (and in-car fighting) that would occur with trying to find a parking spot in the towns and villages, especially hampered by one way streets. It would border on impossible. This alone would be reason to NOT rent a car!
I also think you'll get more of a true Italian feel for the area by using the public transit. You'll be less isolated and living more the way the locals do.
The discussion that if you want a car for a specific day (the one that pops to mind is driving from Ravello to Paestum) is an excellent one. That might be the one day that you might want it.