So what makes regional accreditation so great? Do other countries have it? Why can’t a major regional body go national? Is it the law? It all sounds like a scam from that to textbooks.
National accreditation boards in general, and especially those that are not field-specific, are seen as being less rigorous, faster to accredit, and more profit- or prestige-motivated than the regional accreditation boards. Most are relatively new and some of the most prominent (e.g. ACICS) have been involved in specific controversy over the quality of their work. Because the regional accrediting boards are older and better respected, the first question about a university that advertises a national accreditation tends to be "why aren't they regionally accredited?". It is both symptom and cause of this difference that most for-profit and otherwise "questionable" institutions are nationally accredited and not regionally accredited.
While some national accreditation bodies are considered rigorous and respectable (e.g. ABET), they tend to be field-specific (in the case of ABET, engineering) and often require regional accreditation as a prerequisite to accredit a university, as they consider regional accreditation to be the indication that the broader learning institution is able to provide a quality program. Usually specific departments or colleges of a university will go to these types of accreditation boards to add a credential to specific programs (e.g. "ABET accredited computer science program") above and beyond the university already having a regional accreditation.
There are schools with a shit curriculum that have regional accreditation. The reason it matters to Americans is because most large universities are more significantly more accepting of transferring credits that come from a regionally accredited school.
I've been to nationally and regionally accredited schools. My degree was from a top 200 school in the US. The classes I took at the "lesser" institution were significantly more rigorous and significantly more effective at teaching skills relevant to the major I was studying.
I've known several people that went to schools that have been sued or otherwise have terrible reputations for being predatory for profit schools like ITT and the University of Phoenix. Their coursework was frequently more difficult than mine.
To be fair though, those schools were sued partly because of their shady high pressure sales tactics and for tricking people into taking on massive amounts of debt to take classes. That's pretty scummy, but based on the experiences of my friends it seemed like they at least had legitimate work to do in order to graduate.
Note: I don't doubt that on average the regionally accredited schools have been more thoroughly vetted. I'm just saying accreditation is not even close to perfect so the only reason it matters in the US is because the credits are easily transferable.