You make the assumption that she was captured because she was a Uzumaki, there are lots of other assumptions that could justify this capture.I won't try to say that one of those assumptions is better, just that it exists.
I explained why Uzumaki should be a safe name compared to Namikaze when Naruto was born.
Your assumptions aren't impossible but let me make assumptions and tell me if you can prove that they are impossible.
There is nothing in the manga that says that Mito wasn't put in a cave like Bunpuku just after she became a jinchuriki until her death, making her existence completely unknown by almost everyone.
Kushina wasn't really kept as a secret, but the fact that she was a jinchuriki was, the fact that she was a very powerfull kunoichi may be unknown by the other villages, even by the average villagers of Konoha.
Her goal was to keep Kurama sealed in her, she may just has lived a simple life of a simple housewife of the Hokage.
It's not even said that she was known as Kushina Uzumaki, like Tsunade she could be known as Kushina.
Let's say almost nobody knew that Mito and Kushina were jinchurikis, that they were Uzumaki and even that they were kunoichis.
Her class mates gave her a nickname, it's not said that she was known around the world with that nickname, let's say she wasn't.
She is the Hokage's wife, she stays at home doing the housewife so that she can keep Kurama under control while she is under protection, she never goes out of the village.
I agree to the fact that another name would have been the best solution.
Let me combine my assumtions like I let you combine yours (an outsider+a surname that almost nobody eared about for decades or no surname at all+a nickname given and known by her classmates when she was young+the hokage's housewife), so she is a simple housewife of a Hokage, she isn't known to have ended a war, almost nobody remember her existence and her surname, like nobody remember the surname or the existence of Mary Todd Lincoln.