To rent a car and, at the same time, to travel on public transportation will become a costly Switzerland visit. For me, the first step to being more profitable is to forget the rental car and go all the way by public transportation.
And, most importantly, some of the places you mention, such as Jungfrau, Schilthorn, and Matterhorn (Zermatt) are all care-free places. You cannot go there by car, it’s public transport and cable cars only. Zurich is an anti-car city; parking is difficult to find, and when found, the cost is extremely; many inhabitants of Zurich do not even own a car for this reason. Same when visiting the mountain places, you have to park your car, pay for it, and in addition, still pay for the public transportation to take you further up.
There’s more to trains than the cost of a ticket. There are no parking fees, no high petrol costs, no wasted time getting lost, stuck in gridlock traffic, fines for speeding or for not knowing the country specific traffic code. Furthermore, there’s also the ease and relaxation of train travel, where you sit back and enjoy being transported through the beautiful countrysides or mountain peaks. Take a picnic lunch with you on the train, for example, while enjoying the scenery.
I am not saying this all because we are the SBB community; I am saying this because I am an experienced traveller. I encourage you to do the same as thousands of other tourists visiting our country every year: try out our public transportation, you will be amazed how easy it is to use.
Now, concerning whether the Swiss Half Fare Card or the Swiss Travel Pass is better depends on your itinerary. The Swiss Travel Pass might be a good deal for what you have mentioned as itinerary above. Regardless what ticket, for the mountain visits you still need an additional ticket as the Swiss Travel Pass and the Swiss Half Fare Card offer a reduction only.
If you want to have your itinerary validated in detail, I suggest making a post in the TripAdvisor’s Switzerland Forum or on myswissalps.com. Both travel platforms are excellent for travel advice for Switzerland travel. TripAdvisor has more than 20 local destination experts that provide voluntarily help.
Come back to us if you have more questions concerning the tickets, we help you with pleasure.
Ps. Our English-speaking community you can find here: https://community.sbb.ch/?category.id=sbb_en&profile.language=en-gb