I would suggest you find an attorney to do the paperwork for you. The attorney should be in the state you want to register your business. Your attorney can advise you further.
with that said …
you can open up the LLC in whichever state you want - even remotely (search.google.com is your friend). Each state has website(s) for corporate business registration(s). You should be able to do it on-line for most states and pay the filing fees. However, some states (e.g. MA) limit access to incorporation website for traffic coming only from the US (and may be canada?). In all cases, registration will require a US physical address in the state you incorporate. If you have an attorney or reliable contact then that address will suffice (no PO Box address).
You will need an EIN (tax) number from the IRS. you can request it on-line or your attorneey can do it for you. In the EIN request form, you will have to specify the type of company. From your description, a C-corp may more more sense than an S-corp, Inc, and other(s) for US taxes. You should look into this more, based on your situation. Your attorney can advise.
Depending on the state you register in, there may be state and perhaps even local (gemeinde/city?) taxes involved. you might need to get a state TX ID also. Your attorney or accountant can advise you.
As MedeaFleecestealer said, you need to check with Swiss banks for your banking needs.
Regrettably, no US bank will open account for you, until you are physically present there. Even your attorney, with your power of attorney, will not be able do it. Most US banks will ask for physical presence and two forms of ID to open accounts. Perhaps also additional paperwork to indicate a legitimate business venture.
Getting any credit from US banks, even credit cards, will require 2 years of building credit, starting with a prepaid credit card with a miniscule credit limit.
May be look at Wise.com business account? Wise.com may do that for you, but you have to check. Wise.com does not send out physical checks, even via on-line banking. The US is still somewhat backwards where many normal financial transactions must be made via checks. For example, IRS will only accept paper checks or want your US account details to draft directly from/. I’m not aware that IRS will give you an account number for you to send them payment. Same with many other businesses. They will want a debit or credit card instead. Many of them will not accept foreign issued credit/debit card - card must be issued from a US based financial institution.
If you are mostly remote, then many businesses will not want to deal with you - accountants for example.
I assume you want to keep your CH C-permit. Regrettably, the US system is set up and specifically geared to people who are living in the US. Doing this remotely, you will run into additional challenges …
In general, there is a lot of ‘fraud’ in the US. For this reason specifically, people and companies, in an effort to protect themselves, simply make it complicated, insist on US based banks/cards/payments and push the responsibility to you as if it is ‘your problem.’
All the best …