It's not something Burmese women are used to doing, and if you think about it it's kind of a weird holdover from a much more patriarchal time. But, if she'd known how difficult it would be I think she would have anyway.
I imagine it would be extremely difficult to operate in the US without a last name. She doesn't have to adopt yours obviously, but she could pick one.
For example Osama Bin Laden. Bin is not his middle name, and Laden is not his last name. It means Osama son of Laden. But his relatives in the US use "BinLaden" as if it were a last name.
Or, as the article says, "39. People whose names break my system are weird outliers. They should have had solid, acceptable names, like 田中太郎."
Sometimes you just gotta make accommodations to where you live. Having a last name for legal purposes doesn't really change anything about you.
If email systems were a national thing, and I'd move somewhere where they didn't accept "+" in email addresses for, say, official government communications, then .. yeah, I'd have to get a new email address.
Personally I find it much more annoying if people mispronounce my name (which happens in a personal face-to-face setting) than if I had to write my name down in a certain way on a form, to make it work with the system. It's just an identifier for the system. I wouldn't get the same SSN or phone number either--which is just about as silly to expect.
It would be nice if they had standard ways of dealing with names that do not fit in such a system, so that at least the variant to make it work would be the same everywhere else. But to demand it to be taken verbatim and work correctly in the system, that's like moving to another country and demanding you keep the same land line number.