The US has plenty of people looking for remote jobs, not much incentive to hire from overseas unless the employer needs very specialized skills not available in the US. Sometimes the salary differentials make it worthwhile to hire from overseas, but that differential has to get large enough to make up for the hassle. That offshored outsourcing business mainly revolves around larger agencies that have a lot of employees hired out to US companies.
Companies don't hire corporations as employees. For freelancing you might want a legal business entity, but freelancers generally don't create tax/legal/accounting problems for their clients regardless of where the freelancer lives, so not really a concern. Besides timezones and questions about taxes and visas, companies don't want to have to learn about and comply with foreign labor laws, which they have to do if they start hiring employees in other countries. Employers who do have experience hiring people from other countries may have experienced language and cultural differences that make them hesitate.
If you want to work remote for US companies while living in Europe freelancing is the simplest path, with the fewest issues for customers to object to. You don't need a US business entity. Taxes remain your problem as a freelancer, not the customers'.
I have worked remotely for US companies for over a decade. I probably have an easier time finding US customers because I have US citizenship and speak American English, but I think contacts and work experience in US businesses counts for a lot more.
