> What does the "particle" model bring to the table that a localized (wavelength-sized) wave/vibration could not?

A lot of the article is about this. Start with the section "The Wave Function of Two Particles and a Single Door". The wave packet view can't explain why you don't for example see a "particle" (that is, a dot on a detector) show up simultaneously having gone through two different doors. You have to think about it in terms of a wave in the space of possible joint particle positions.