I.M.E. Broken home is usually a layman’s term as in divorced or separated parents when describing the children’s background.
Nowadays they use the word “dysfunctional” which basically includes any kind of unhealthiness, whether the parents are together or not. Can refer to abuse going on, neglect, or just “tzifoorenkeit”.
Now about the second class issue – that is unfortunately a result of stigmas in some communities that are absolutely no fault of the children of course. My view on why this happened is that When people used to hide most things deemed “imperfect” on the street, (special needs children, disease, mental illness, etc.) the blemishes that were obvious to the public appeared worse in their eyes (due to the denial factor) & shunned even more.