This seems sort of interesting as a sort of emergency backup, but it seems like the real solution for any sort of long distance/long duration in space is just making mass to orbit dramatically cheaper and shielding the spacecraft.
“It still gets people surprised,” Milstein says. “Everybody asks, what about the head? But we’re actually able to reduce the effective dose by 60 percent without protecting the head, the arms, or even the legs.”
Whilst I understand the maths here, I can see why someone would be nervous!
You need to protect anything with relatively fast dividing cells best because this is where the cancer risk is highest. The brain with its slow dividing cells and fast killing tumors is the last thing to protect.
The difficulty is making it protective but light enough to not significantly hinder the spacecraft or person wearing it, all of which is covered in the article.
Poor design, flawed assumptions, manufacturing defects, inability to bear the gravitational and rocket launch forces, inability of the jacket to last the duration of the radiation exposure - these are some reasons that come to my mind on why a radiation-blocking vest wouldn't have worked for the journey to the moon and back (i.e. your question "Why wouldn't it?".)
Whilst I understand the maths here, I can see why someone would be nervous!