I've looked at your revised itinerary. Thanks for avoiding the "all capitals."
I'm still not quite sure which days you would use a car and which days you would use trains. Certainly, the Golden Pass route is trains. With that uncertainty, i have not checked your itinerary vs. travel times by either mode.
My quick impression is that you will spend a lot of time traveling, as opposed to sightseeing. When you have a lot of people, it is certainly cheaper to rent a car, but you'll be working hard at driving, and going by train lets you see the sights without distraction while you travel.
In particular, coming north from Lugano over the Gotthard Pass route is one of the most scenic train rides in Switzerland. Going by car has you driving quit a long distance through a tunnel which I particularly dislike. Two way traffic. Since you mention Wilhelm Tell, you may be planing to do that by train and boat anyway.
What you want to see and do is very much a personal judgement. Mine is influenced by the fact that I live in the USA and have seen Niagara Falls many times. So, If i were following your itinerary, especially if i was tired of spending time in a car or on trains, I'd not go to to the Rhine Falls on May 16th, but would rather spend more time in and around Luzern, or possibly substitute Bern for the Rhine Falls. It is somewhat closer to Luzern by train. Direct trains once per hour on the hour take 60 minutes....in between are trains which take up to 90 minutes. Schloss Laufen am Rheinfall ( station) is an hour and 40 to an hour and 50 minutes from Luzern.
I'm uncertain about the amount of walking required at the falls. When we went there many years ago, we perhaps went by a different route or to a different viewing site, and I recall some walking.
In Bern, my quick trip is to walk from the station 20-30 minutes downhill to the Nydegg Bridge through the scenic old town, cross the bridge, look at the bears below, and then take the short steep walkway a short distance up hill to the Rose Garden. Great views, nice flowers, restaurant. Walk back down to the bridge (restaurants) and catch the tram in front of the Brasserie Bärengraben ( I think it is the #12 tram) back up through the old town to the station. Or, walk back to the station, if you wish. Approximately 2 hours......can be less.
There is a tram route with changes that can bring you from thestation to the "back side" of the Rose Garden, but the old town deservesa walk. Or, you can use it to return t the main station, although it will take a bit longer than the direct route through the old town.
Since you are staying only a short time near the Jungfrau, and thus may find that clouds and weather convert the ride up high to see the sights into only a disappointing journey to see the inside of clouds, Bern is a nice contingency plan. Many travelers try to plan a couple of days in the Jungfrau region, specially in Spring, because the weather is so variable. It changes a lot from day to day, and that way you have a better chance to get clear weather.