You would have to work as ANobAG, which means the ball for paying Swiss social security will rest in your court. You pay the contribution for the child benefits scheme and the compensation office’s administrative fee, which are both normally covered by your employer. If you want to use the pillar 2, you will pretty much be limited to the Substitute Occupational Benefits Institution, with you covering the full contributions, but pillar 3a might make more sense in this case. You will have to get accident insurance from your health insurance and get private loss-of-income insurance if you want to match the accident insurance you get through Swiss employers.
It’s pretty standard for US and even Asian companies to pay salaries in US$, with you carrying the currency risk. I did this for some time.
You can’t really claim anything from your US employer from a legal perspective - not even paid vacation or sick leave. Your US employer is not subject to Swiss law. But of course you could negotiate paid holidays, a private pension fund and compensation for your Swiss social security contributions, etc.
Ultimately it all comes down to gross and net salaries. If you can cover all your bases and still come out with more, then why not?