Even a “legal name” can be different things in the same legal jurisdiction, or not be a concept that really exists at all. I’m in the USA and my birth certificate doesn’t have distinct first, middle, and last name fields. It’s just a single line with my whole name typed out.

That's sort of my point. The Legal Name field is a single field, like it apparently is on your birth certificate. It's just a byte-stuffer identifier filled with something that can be used by the business for any legal forms that come up.

The individual parts inside don't, and shouldn't, matter.

Also, long-standing common law precedent in the US is that people can call themselves whatever they like for any reason at any time as long as it's not with intent to deceive or defraud. This is the basic principle behind, for example, businesses run using stage or pen names (see: "Prince").

Even stranger is that birth certificates are the foundation for all identity in the United States, yet there are >14,000 forms and very poor standards around them.

Well, the number of middle initials/names that I used in the past is dependent on the constraints of the particular system, so I have no possible way to be consistent going forward. I would rather not use any at all, but Social Security and credit cards have one, while my credit union has the other, the DMV and my birth certificate have both, and I use neither when it doesn't seem important. It's definitely unreasonable to use my full legal name for everything, so I think at least three would be needed.

So long as the business gets something that they can use as a legal identifier, they don't need to care about what is inside it.

The whole problem of people not having middle or family names goes away when it's treated as a single field. You don't care about tokenizing the field, because you only ever reproduce that field verbatim.

If the user believes a middle name identifier isn't necessary, then it probably isn't. It's only one aspect of the identifiable information used to identify the legal entity of that person anyway. If you have a name, address, possibly CC information, and email, that's probably more than enough to claim they're a customer for whatever legal purpose the business needs.