Well, the combination of a rental car and public transport tickets is the most expensive way of travelling in Switzerland. Car rental, gas, and parking are expensive here, so is public transportation. You should choose either or, while many interesting places are not accessible by car at all, only by public transportation. For example, all mountain tops and view points in the Jungfrau area are not accessible by car, some places are even so-called car-free villages, such as Wengen. And mountain railways and cable cars can be very expensive without a Half Fare Card.
Unfortunately, Adelboden is not conveniently situated to explore the area, although the village is charming and very Swiss.
No, you can also buy Saver Day Passes without Half Fare Card, starting from CHF 59. But, they are not valid on most mountain railways and cable cars. Also Supersaver tickets can be bought without a Hald Fare Card, at a higher cost than with Half Fare Card, of course.
In my opinion, you should opt for the Swiss Travel Pass or the Swiss Half Fare Card, or the Berner Oberland Pass. The first and the second offer the complimentary Swiss Family Card, with it, children aged 6-15.99 travel for free when accompanied by their parents, and they hold a valid ticket.
Swiss Travel Pass – the All-in-one-Ticket for Switzerland | SBB
Swiss Half Fare Card – Switzerland at half the price | SBB
Ticket for the Bernese Oberland / Berner Oberland Pass (berneseoberlandpass.ch)