score:5
Will I get a better trip (in terms of peace of mind and other travel related troubles) if I move around by train instead of by car?
Yes.
You seek to visit Zermatt, Lugano, and Verbier.
Zermatt and Verbier are car-free. Switzerland has excellent (if expensive) public transportation. In most areas, you do not need a car.
If you want to explore the area around Lugano you might rent a car, as certainly it adds flexibility, but then I would recommend only renting a car in Lugano and returning it there, while keeping the rest by train. Lugano is much warmer than Zermatt and Verbier anyway, so you will likely not have to drive on snow unless you go high up into the mountains.