> In order to do that, your LLC has to elect to be taxed as an S-corp, which requires it to file its own return
I think your interpretation is wrong and you have stripped the context on your IRS advice you quoted, changing the meaning. At the start of the paragraph containing your quote, it says "For federal income tax purposes, a single-member LLC classified as a disregarded entity generally must use the owner's social security number (SSN) or employer identification number (EIN) for all information returns and reporting related to income tax."
IMO, this has has to do with how one files their income and/or report to the IRS - as a pass-through entity, the IRS expects the human behind the single-member to file the income under their own own SSN even as after are distributing invoices with their LLC's EIN. IANA(Tax)L