Is this a real rule or a way of selling me American lawyers?
It is a real rule, and the citations are below so you can read it yourself rather than take our word for it.
It is also narrower than it sounds. It governs who may represent an applicant before the Office. It does not govern who may read your document, explain it, or help you decide what to do.
Who it applies to
It turns on the domicile of the applicant — normally the company or individual that owns the application.
A US-domiciled applicant is unaffected, even where the inventors are foreign nationals.
A foreign-domiciled applicant is affected, even where the inventors live in the United States.
Being represented by a foreign patent attorney does not satisfy it unless that person is on the USPTO register.
How to comply without disrupting your existing advisers
The usual arrangement is that a US-registered attorney or agent is appointed of record and files, while your existing adviser continues to run strategy and hold the client relationship. This is routine work for both sides and does not require you to move the portfolio.
What causes trouble is discovering the requirement in the last week of a reply period. If your application is being prosecuted from outside the United States without a registered representative, resolve it before the next deadline arrives.