Its probably more complicated than that. In areas that don't have bike lanes, people tend to just bike on the sidewalks, and pedestrians just put up with having to move out of the way. The reason is likely a mix between being historically ignored by the city, not bothering to really try, and also just knowing that sidewalks are safer to bike on (even if inconvenient).

When wealthier people move into a neighborhood, expecting bike lanes, they push for them. They probably have more power/influence, but they also have a higher since of entitlement that causes them to not give up, even if the city pushes back.