In the US, you actually have to register your business as a foreign entity in every state you operate in (foreign in this context means “out-of-state”) and it’s a minor annoyance, it can and does delay business.
Indeed. I'd say a major annoyance. After all it includes filing taxes and sometimes you have to "align" your company name if for example you apply for federal grants.
In fact the best way for the EU to help start-ups is to stay out of the way as much as possible (a big ask...)