For me the simplest way to work for from EU to a USA company would be to register a mini-company for myself. I don't have one, but many freelancers do it just for local customers, because you can get a bit of tax savings that way.
The process then is simply making a periodic invoice from my company to yours; the ongoing costs involved would be whatever an international money tranfer costs for you, + I'd pay something like $50/mth to a local accounting company to handle all the taxes&reporting for that.
I don't know of any full "remote employment as a service", but there are a lot of outsourced accounting companies who'd do the taxes & paperwork - most local small businesses do it this way, since there's not enough work to have a full-time accountant/finance person on board. Actually executing payments and managing days off is a bit trickier, since the first requires full access to the company funds, and the second involves managing people, which is hard to outsource - you probably could get them to calculate 'what is owed to whom' by the local laws, but you'd have to pay salaries yourself & schedule vacations directly with the employees.
This however doesn't grant you days off - if you don't work you don't get paid.
Also, at least in Germany, if you only have one client/customer, you're vulnerable to committing a tax/social security offence called "Scheinselbstständigkeit" (roughly: faked entrepneurship), as you'll end up paying less in taxes, social security and insurance than a full-blown employee and people have abused this in the past to maximize their profits on cost of the employee or the systems.
Sure, this is the 'contractor' approach where you bill for services provided at a higher rate than a pure salary, and pay for your vacations/sicknesses/whatever out of that.
As for that "Scheinselbstständigkeit", I believe most other countries have some similar legislation. It's not an issue with freelancers as you have rotating clients, even if it's only one at a time; but yes, if you're creating it just for a single employment then that can be treated as tax avoidance - having the employer run a shell-subsidiary-branch would be preferred in that case.
The process then is simply making a periodic invoice from my company to yours; the ongoing costs involved would be whatever an international money tranfer costs for you, + I'd pay something like $50/mth to a local accounting company to handle all the taxes&reporting for that.
I don't know of any full "remote employment as a service", but there are a lot of outsourced accounting companies who'd do the taxes & paperwork - most local small businesses do it this way, since there's not enough work to have a full-time accountant/finance person on board. Actually executing payments and managing days off is a bit trickier, since the first requires full access to the company funds, and the second involves managing people, which is hard to outsource - you probably could get them to calculate 'what is owed to whom' by the local laws, but you'd have to pay salaries yourself & schedule vacations directly with the employees.