I lived in Detroit next to a major corridor (Cass Ave) that got protected bike lanes going both ways and can confirm, it was _awesome_. The biggest differentiator I noticed is that the bar for how "hard core" of a biker you had to be to use the infrastructure went way down. You started to see many more families and women bike than you saw before.
The problem I repeatedly see with public opinion on both bike infrastructure and public transit is that both benefit and suffer from network effects.
When the network is already large, adding a new connection is amazing: the whole network gets access to that new node, and the new node gets access to the whole network.
When the network is tiny - as it is in most American cities - adding a new connection looks pitiful: the connection often looks like a "road to nowhere" that does nothing to solve the broader problem.
I think that one thing that can help this is communicating any improvements within the context of a broader plan, e.g. "This bike lane connects an additional 3,000 residents to the downtown business district. More importantly, it will serve as a central artery in connecting an additional 20,000 residents over the next 5 years."