Does it make sense to rent a car to tour the Amalfi coast then Rome?

There are quite a lot of places in Europe that don't have train service, often because of terrain. My solution is to take a bus.

Whether renting a car for the Amalfi area is reasonable or borderline insane may depend on the time of year (based on what I remember of earlier comments here). When do you plan to travel?

No, a car is a nuisance, traffic is terrible, parking hard to find. Rome to Naples, train. Then Circumvesuviana( commuter) train to Sorrento, ferry or bus to Amalfi Coast. Or driver service from Naples to Positano. Or train to Salerno, ferry to AC.

Well, the rental car return locations are in either Sorrento or Salerno. So some form of public transportation to return the car is necessary. While we do drive the AC, it's in early may/mid October. If your planning on high season, you can't go any faster than the busses in front of you. Parking is limited and could be expensive and very frustrating.

There are also buses and ferries from Salerno, however.

You can reach Salerno by high speed train from Rome. Buses stop in front of the Salerno train station, ferries are a couple of short blocks from it.

You can also rent cars in Salerno, instead of Rome airport, if you prefer a car. Rental agencies are also near the Salerno station. Taking a train to Salerno would be my recommendation if you plan to go to the Amalfi Coast immediately after a transatlantic flight. Driving for 3 to 4 hours from Rome airport to the Amalfi Coast while jet lagged would not be advisable. You can rest on the train. Then once in Salerno, if you plan to drive, at least the drive is just over one hour.

We took a train from Rome to Naples and the Circumvesuviana to Sorrento, where we spent a couple of days. We rented a car (the smallest we could get) there and drove the Amalfi coast, spending one night and then continuing to Paestum. Another night after that, and we returned the car in Sorrento before taking a hydrofoil to Naples for the last leg of our vacation. That worked fine for us, but driving that road may not be for the faint of heart. My husband is not in that category and the only thing he liked better than the trip one way was that he got to do it again on the way back. Our trip was in early October, not at the height of crowds, but still plenty crowded on the road. If you are willing to do the drive, that has the most flexibility. A car with a driver may be next best, because if you take a bus and don't get on the sea side, you miss the wonderful views.

You can definitely reach Positano other ways than a cab.
The SITA bus is very inexpensive and goes there for example, as do other busses and ferries.

There could be some benefit to having a car though ; would definitely make a Pompeii daytrip easier from Positano as one example of a benefit ; as long as you understand the negatives.

1.) Driving in Naples itself should definitely be avoided ;
2.) Parking along the AC is a major problem. Even if you have parking where you are staying on the Amalfi Coast ; each time you use the car to go somewhere you may not find parking available. This makes the car kind of pointless to visiting the towns along the coast.
3) The coastal road is a challenging drive: Traffic backups, scary situations that happen on the coastal road when an approaching bus meets you head on at a narrow stretch of road ; will you be comfortable reversing in such a situation to a wider stretch of road so the bus can pass? Aggressive driving locals on scooters are a aggravation as well.
The driver definitely won't have time to enjoy the scenery.