OK here goes.
1. - expats from UK and USA or other Anglo-Saxon countries are mainly in those 2 large towns, Geneva and Zurich and the smaller financial services/lox tax, Zug. Many have a 'parrallel' life-style, never learn the language, and spend most of their time hanging around together. One reason is that they are often sent by their company for a limited number of years and do not come to live here permanently. As the Swiss and European school system is so different- especially at 16+ (UK with 4 subjects in Year 12, and 3 in Year 13- whereas here and in Europe in general, the 16+ system remains very broad up to the Baccalaureate at 18/19)- expats prefer to educate their kids separately, studying for GCSE's and preparing for A'Levels. Even at secondary school- kids are split into vocational and academic, based on all subjects- but with one main subject being the local language, eg German for Zurich and Zug, French for Geneva. And that means that kids arriving after, say, the age of 8, do not have time to acquire the necessary fluency in spoken, let alone written language before that separation takes place. In the UK for instance- kids can continue into a very narrow A'level system, even if they do not achieve C grade in English and maths- and then take those GCSE's again alongside AS and As. They can also choose subjects which are easier, or less demanding from the language point of view- be it IT, Design, a foreingn language they already have a good grounding or are fluent in, etc.- which is just not possible here- where all main subjects have to be studied by all, and the exam passed as a whole (fail one subject and re-take all).
Many companies also offer to pay for private education for expats, as a carrot.
2.- no private schools are not illegal (some Counties do make it illegal to home-school, but not all do- you have to be qualified to do so though). But only a tiny minority choose to send their kids to private school. Again, more in Zurich and Geneva. In my County- the % is absolutely tiny. A private British school was opened a couple of years back, as there has been a big increase in foreign firms here- but the take up was so small, they closed. In my region- we have many multi-millionaires who are the CEOs and owners of prestige watch making firms and other businesses too- as of course solicitors, doctors, architects, etc, etc,- they all send their kids to the local school. Because the local schools are great, well funded, etc- and because they make sure they use their influence that continues. They actually believe an excellent education system benefits all - including their own- and I can assure you NONE of them are socialists or communists.
3.- no... some expats do. The only 2 kids I knew in my day who went to Public Boarding (very famous) Public School in the UK were my cousins- whose parents worked for the United Nations in Geneva (she Swiss, he British, having gone to same school). Why would they do- when the local schools are great?