Yes, it is exactly that: supply and demand. Many more applicants than flats available.
Here are some ideas to help:
Tell everyone at work that you are looking for an apartment. Your manager, your colleagues, the HR people, the staff in the canteen, the cleaners, the receptionist, the caretaker. Give them your name, phone number and mail address (there and then, on a card) and ask the to let you know if they hear of anything.
As you and your wife make contacts outside of your work, tell those people, too.
Design a "looking for a flat" flyer. Ask your most trusted work colleague to check it to ensure it doesn't have any serious cultural errors. Say something like: "We're a reliable, easy-going family (m aged 39, w aged 36, son 4, son 2), learning German, looking for a home in this area. Ideally 4 rooms, 3 would do, too. We'd be grateful for the contact details of anyone who knows of a rental opportunity near here. Thank you very much.". Go for lots of walks. Whenever you find a street in which you think you'd like to live, and you can estimate the rentals based on the ads you've seen (or the flats you've already applied for), put a flyer in each letter-box.
Put up an ad in the local supermarkets. Usually, you have to ask them for their pre-printed post-card sized form, and put your info on that, and then you are allowed to put it up on their noticeboard. Smaller shops may allow you to use your flyer.
In doing so, take care not to moan negatively about the Swiss system. I really liked your thread title, because you do not say "things are stupid here", but instead, with "looking for options" you are asking for advice and suggestions. Therefore, I'm sure you will be able to apply the same to your flat-search. Tell people how pleased you are to be working here, how amazingly beautiful you've found the lake, how glad you are of the excellent public transport... and that now you're looking for a flat to call your family home.
Many rentals are found though these kinds of personal conversations, rather than just through the ads published online.
I wish you all the best.