If I were in your position, I think I would try to take one of the following three paths.
OPTION 1 Focus all my energy on what is likely to improve my chances of getting to Switzerland, and try to expend as little energy as possible railing against the Swiss system (as understandably annoying as you find it). Keep up your German, and work diligently to improve it. Find out what it will take to ensure that your permit to stay in Turkey is renewed for the max time possible, and fulfil those requirements (so you don't have to worry about having to leave Turkey. And also so that you don't ever risk being anywhere illegally, because having been illegal will pretty much preclude you from ever getting a permit to stay in Switzerland or an EU country). Find out what it would take to get permanent residence in Turkey (so you can breathe more freely, having a secure base). If you haven't already done so (but I presume you have) reduce your living costs in Turkey to the minumum. Research each and every EU country thoroughly to find out about the requirements to enter the country, to work there, to get permanent residence and to naturalise, in some years from now. Choose the most advantageous four or five and research which kinds of skills they need. While you are in Turkey (or back in the US, if you have had to leave Turkey to avoid being illegal) train/study/learn/practice to become one of those needed people. Apply to zillions of jobs. Build up a network by becoming known on local fora, per country, specifically dealing with those "needed" professions and skills. Become the interested, interesting, helpful contributor that anyone would like to have as their working colleague. Through that, try to build up personal connections, in addition to the annonymity of fora. Join couchsurfing (it's free) and put thought into your profile, and host people where you stay now (no matter how humble, as long as you are allowed to do so) or at least meet travellers for coffee. Collect good references. Go couchsurfing locally to where you live now in Turkey. Collect more good references. Be shown to be the kind, helpful person that any other host would like to have come and stay with them for a few days. Practice cooking a few good meals so you can contribute whenever you are hosted. Work through your budget, and see whether you can make it to visit your son in Switzerland (perhaps more do-able if you can couchsurf for at least some days or weeks at various people's homes, or in exchange for helping them with something) AND ALSO to visit some of your fora contacts in EU countries, especially any who think their employers might be hiring, and there, too, keep your costs down by couchsurfing. Tell everyone you know, including your couchsurfing hosts, that you are looking for work, and of all the skills you have. Keep good documentation of everything. I think that going that route would probably take you several years to get into the EU, and then you can clock up the 5 or so years it takes to become a cititzen. During that time, you can visit your son and, when he is older, perhaps his mother will let him visit you.
OPTION 2
The alternative is to go back to the USA, to become super high-qualified in a niche Switzerland needs, and apply for a job. After all, many non-EU people do, in fact, get permits to work here.
OPTION 3
The other route I see is for you to go back to the USA and build a full life there, and when you can be shown to be a solid, reliable, established kind of person, appeal to the Swiss authorities (if you have not already done so) to grant you specified, longer access to your child during school holidays, given that you will not be able to visit every second weekend.
Whichever of those routes (or others of which I may not have thought), maintain contact with your son, write him letters, send him photos, tell him you are working on ways to see him, but DO NOT tell him negative things about the Swiss permit system nor about his mother, nor how very hard it is for you to get to see him, nor how much you are sacrificing, etc. I think working 70 hours a week is no joke, so I acknowledge that. I just think your boy does not need to know that, because it won't contribute anything to his well-being. His well-being is served by staying with his mother and knowing that you are present in his heart, and love him, and that you live far away and are working, and that you save money so you can visit him. It may be hard for you to limit it to that (I don't know, though, maybe you already do) but I believe that's probably enough information for him.
Whichever way you go, it's going to be a long journey, and I wish you well!