https://community.ricksteves.com/travel-forum/switzerland/teens-in-switzerland
I see your planning is evolving, and I think two locations will serve you nicely. Not sure why a travel agent would recommend car service, since the Swiss have a world- class rail system. Make sure this travel agent really knows the area.
Definitely stay IN the Alps, in Murren , Wengen, or even Lauterbrunen. Book directly with the lodging for best prices. Your group of five will be a little hard to accommodate , so if you find a great place, book asap, as you're a bit late to secure lodging for summer.
That being said, you all will have a great trip in this beautiful country. The cogwheel trains, gondolas, cable cars, parasailing etc give Disney- like adventure experiences in the real world. Weather is always a factor in the Alps, so you may not get to all the summits that are in your plans. If you have a clear day, go for it, as the next day could be very different.
In Lucerne, I love the open air Cabrio cable car at Mt Stanserhorn. Since it's not in the RS Guide, this doesn't get much press here, but the open air experience is a thrill. Go up for the sunset, it's 45 minutes from Lucerne to the summit (train, cogwheel, and cable car journey.)
If you have the STP, both Mt Rigi and Mt Stanserhorn trips are completely covered. Folks ask if they should explore the summits around Lucerne if they are going to the BO? My answer is yes, as each peak has its own personality and vistas, and you never know when the weather turns , so if you have a beautiful day, maximize your mountain opportunities.
Have a memorable trip!