<<"The simple solution is probably to stay in Zuriuch for a few days and do a day trip to the region.">>
I love Zürich. But, I have friends there, and I go for restauarants, museums, cultural events...not for scenery, even though the old town is nice to walk through.
Sme pictures attached. By the way, don't ever believe that the Swiss don't have a sense of humor. The picture "Concert" was taken as I passed through Zürich main station one day. "The worlds largest Pan-flute concert" was happening on bleachers set up in the main station hall. Fun. And funny.
Plusses and minuses...
If you stay in Zürich with a rental car you have to pay a whole lot of money to park it. 30-40CHF overnight. Few locations. And, the drive to/from the car rental places can be difficult....one way streets, confusing street "grid" ( if you can call it that). Zürich hotel prices are not cheap.
On the other hand, even with good train and bus service in the region near Gippingen, you tend to be limited to trains/buses once per hour, sometimes the buses take midday break.
When I go to Switzerland from the USA, I fly into Zürich, bite the bullet ( cost) and stay in town for a couple of days for jet lag using trains to make day trips....even as far as Geneva once or twice ( 3 hour ride) for lunch with a friend.
I have stayed several times per year at the Hotel Adler for 12-15 years, so I am "ein Stammgast" ( a regular guest) and usually can get an early room on arrival day, and they'll squeeze me into the very busy restaurant.
By Zürich standards, not overpriced.
Then, I typically rent a car, drive to the hotel to get my bags, and drive to a base on a train line in a small village, and stay at an inn where I know the innkeepers. Cheaper, no less friendly, but in a rural or market town. Then, I use both the car and the trains for day trips.
Train to the larger cities. Car to some trail heads with nowadays infrequent bus service.
I spent many years memorizing the routes to/from the Avis office and the hotel in Zürich. The twists and turns come up very quickly, in heavy traffic. Errors lead you astray quickly, so I always carry a smartphone with a map. That luxury did not exist when I first developed that pattern.
At first, I used the frequent trams ( nuisance to haul luggage) or a cab ( quite expensive for a very short ride.)
Using a car in Zürich is not a good idea. ;-)
Renting at the airport...logical for you...adds a surcharge to the cost of the whole rental. Last I looked about 7-10%. That still might be a good idea, if you stay north of there.
With your goals, I'd find a place to stay outside Zürich, maybe north of town, and use a rental car for the villages for a day or 3. Train to Zürich; try not to miss Luzern.
I have rented in Aarau, at the Avis there a few times. About 600 meters from the train station.
I'm not sure where I would look for an inn in that area near Gippingen. . There might be a Gasthof in a nearby river town.
In the mapsearch.ch map, under points of interests/gastronomy, you can find a place to turn on "accommodation" It does not catch every hotel or inn, but it might help. I also check "restaurant."
In Gippingen, I find something called Landgasthof Weisses Kreuz, under restaurants
I googled it. Been there since 1532. New inn keeper (der Wirt) early 2017 "end of an era"
They have 6 hotel rooms.
Website not was not cooperating to show the rooms for me.
That is the kind of place that I'd almost certainly stay at.