Take my response with a grain of salt. I have been working as an offshore freelance dev but I am not registered as a local LLC or a US LLC. I might pick either US or Canadian LLC if I get paid really well for a permanent contract. But until then I hope to work with individuals, startups and small businesses.
One word answer (Pick One): Legitimacy/Accountability/Trust.
> Is there any reason why you need LLC in America to work with larger and established companies?
Companies need people they can hold accountable for the work they do. If you are an American LLC it means that you are a legitimate business that they can interact through proper legal channels and you are legally liable and held accountable under the contract. They are not going to take you to international court for breach of contract but if you are an American business you are fair game.
Also this allows you to sue client in failure of payment or breach of contracts. You have easy access to proper contract engagement services like escrow, small claims court etc.
With larger business you have higher degrees of authority as you come as a business based in America. It makes taxation and legal fillings easier for them. Money laundering acts, audit, tax fraud, wire fraud etc. have extremely strict laws and have serious consequences for a larger business. I have worked with people who are subjected to FinRa (wallstreet people). I had to file proper paperwork or there is a risk of them getting audited.
> Is there any law or limitation why they wouldn't like to work with e.g. European Self-Employed or LTD companies?
It is very grey. You have to file paperwork and send documentation that you are a legit tax paying business in your local country. You have audits and internal control team in large organization that cross checks every transaction made to individuals outside the company. You have to back it up with proper paperwork. Kind of a hassle for larger companies but if a startup, small business or an individual is hiring you that is no problem.
> Unless maybe when people here word 'contractor' they assume they will have to pay 2x more and that's why they prefer perm-employees? Or is there are any law in US that make it harder for companies to work this way?
There are some hassle but it mostly comes down how bad they want you. For individuals, startups and small business the pros far outweigh the cons. I am affordable, I am culturally on point, I am a decent dev. I submit invoice, I get paid. That's it.
But for larger companies there are some legal hassle. But if they really want you, they will find a way. They can hire you as an employee, or ask you send over paperwork as a business. The recruiter can and will make arrangements to hire you. But often times they could hire locally and not risk getting asked questions about you.
Very long answer but again grain of salt.