I don't see a meaningful distinction between a consolidated parking lot and one that's distributed along the side of the road when it comes to tree and grass cover. For dense urban situations car parking buildings allow more than either surface lots or surface street parking.
Parking lots are mostly used only half of the day and empty otherwise, because typically people drive their car around in a few different parking lots. Streets are useful all day long.
I'm unconvinced removing street parking would allow for smaller streets because that space is also used for bus stops and snow management in winter. Chance are street parking don't take place in itself but merely maximize usage of space that would be used up by the streets anyway.
I don't think we need private parking buildings to park our private cars when not in use, ie, most of the time. We need fewer, always in use, public cars to drive on our already public roads.
someone in the chain just needs to bite the short term bullet to start a cascade of events that will make american cities more liveable
...and housing is too expensive because cities strongly resist increasing density. We have indeed found ourselves in a cycle of dysfunction.