Does it make sense to buy a pass

The usual advise is to get either the Half Fare Card, or the Swiss Travel Pass, and not get a rental car at all. Switzerland is a country where the public transit system is excellent, but expensive if you only use it occasionally.

I agree, the question you should ask is Does it make sense to rent a car?

There is zero reason to rent a car in Switzerland! A rail pass will get you free trains, buses, boat rides, plus museum entry. The mass transit is reliable and safe and gorgeous!

If you are relying on the car to get you to the trains and and lifts in the Luzern area, then I don't see how any pass would pay off.

Travelling by car in Switzerland is fine - I do it frequently but that is because of my unique situation - but there are things you do need to be aware of. Speed is highly regulated and strictly enforced; if your car is coming from a different country you will need a 40 CHF Vignette on the windscreen; parking is expensive and rarely free; the best views are often from the train.

When we were visiting Luzern by rental car we chose to stay at the Hotel Balm in Meggen, which had free parking and included the city pass which gave us free transportation. It was a 10-minute bus ride into Luzern from the stop directly in front of the hotel!
When we visited Mt. Rigi, we drove to the base and parked in the multistory garage in Vitznau. Some hotels in the Luzern area also offer a deal on the Tell Pass https://www.tellpass.ch/en so check into that!

I haven't visited Appenzell so cannot speak to that.

For our upcoming trip in early September we are with 8 friends and are traveling by train. Frankly, even with the Half-Fare card and SuperSaver fares, I found it to be pricier per person than renting a car and a hassle getting all our journeys booked.