After trying to fix this for more than 6 months it's obvious that:
1. There is a problem with the drivers probably (or the hardware, less likely). After trying at least 7 driver versions and getting same results I gave up. Also the newer driver packages come with worse monitoring tools. After becoming (involuntary) a wireless "expert" it's time to move on.
2. The workaround I provided above (keeping the wireless continuosly in use by streaming something) works.
3. There is no real help from Intel. Posting some threads (that sometimes don't even apply to the situation) was not helpful.
4. The vendor of the computer (Asus) gets (the same) newer drivers from Intel from time to time, so no luck with that either.