My advice would be to also consider the addedd things you get with semi private in general, not only for the birth itself.
For example, semi private comes with the complementary insurance which also reimburses osteopathy, acupuncture and massage which are all very helpful during and after delivery.
So it is a matter of comparing the cost but also the overall benefit, not just the delivery itself.
I had semi private (courtesy of my company) and have not upgraded to private myself and I am glad to be looking forward to a delivery in a fully private room at one of the hirslanden kliniks as they are wonderful
Also, do not forget that upgrades are possible BUT only if available. First will come the patients who actually have the semi private or private insurance.
Nevertheless, indeed in CH if we are discussing medical care only, you will get GREAT care in any public hospitals. the private kliniks are more like hotels with medical facilities

Nice but not a must
Lastly, it is true that you can have your complementary with another company BUT be aware that this will create some more hassle as sometimes they will start juggling the ball and say the other company should pay. I have heard this from a number of friends...
And whilst on topic, a must is also not to forget to insure your unborn baby in case you are thinking of having a complementary for him/her as if it is done before the birth there are no questions asked, after the birth it becomes a bit more complicated with long health questionnaires.
Am sure you will make the right choice for yourself and all will go well!
K