While pushed to the streets is grammatically correct and carries the meaning you want to convey, it's a little off idiomatically.
First, it would be pushed onto the street. I have a strong feeling that street should be singular. For example, you say I live on the street and not streets.
Here forced would work better than pushed. For example, "He was forced to the street."
The problem with pushed is that it has a connotation that the agent doing the pushing is a human being and the pushing is literal (i.e. physical). For example, "He pushed me to the street." describes a rough physical act and has nothing to do with being homeless.