I'm only familiar with UK rules on the topic, and of course the UK is no longer in the EU, but here at least the standard for determining employment status is quite complicated. The ability to work for multiple clients is one of the factors, as it speaks to the control and mutuality-of-obligation tests set by IR35, but it would probably not (alone) be enough to determine employment status either way.

> If it was legal to work in the office of your only "client" 40 hours a week on a permanent basis, then any EU company could ignore the entire employment legislation of their real country by setting up a shell subsidiary in the US.

That wouldn't work because it would be an obvious sham designed mainly to avoid the EU company's responsibilities under employment law. Courts see through those shams very quickly.

Technical people -- including me -- like to try and reduce the law to a series of digital if/then/else tests, but reality is much more analogue. If you're one of a small number of highly-experienced remote contractors engaged by a US-based client with no local subsidiary, the authorities are likely to accept the arrangement, or at least not to spend significant amounts of time investigating it. If you're one of very many Uber-driver-like "contractors" working for a company that is obviously dodging its local employment law obligations, then they're much more likely to be interested.