There seems to be a low tolerance to the construction disruption, especially to adjacent businesses and residents. Cut and cover also had high cost of moving infrastructure. I also wonder whether there's some relationship with the speed of construction - in NYC, ripping up the whole street to build a Subway was perhaps more acceptable if it's done in a year or so.
One note: new cut and cover projects tend to actually use cover and cut: first the walls are built underground (often piledriven), then the top is covered with a concrete plate, then the Earth dug away from below. This minimizes the surface disruption.
Most of these tunnels are being built in heavily populated areas where surface space is at a premium. You'd have to tear down thousands of apartments and offices to make any meaningful progress on a cut and cover tunnel inside a city unless the tunnel is exactly following existing roads. That would be both very expensive and far more disruptive than just a little traffic
Besides the physical activities, the number of interested parties that become involved when touching so many things drives up the bureaucracy with each party bringing their own engineering requirements and conditions for cooperation.
It's cultural; cut and cover is more popular outside of the US. The Shanghai Metro (world's longest) is constructed using mostly cut and cover, and in the Anglosphere I remember seeing some of the Sydney metro being dug that way. One part of the Shanghai approach that I like is that they build their tunnels in advance of development, assuming that there will be demand in the future. This avoids a lot of cost and disruption, and I think it's a safe bet - of course the city's growth will follow the preemptive subway lines you built, that land has subway lines.
I wouldn't characterize cut and cover as a mild disruption to car traffic - it's a shorter duration of disruption than tunnel boring, but a block at a time will be completely shut down.
They already shut down blocks at a time to have six men inspect a man hole. Most business traffic in LA at least is from people walking on foot, and chances are you did not park close. Having a little pedestrian bridge over the open cut would minimize impact to local business. There are plenty of parallel arterials that could have temporary street parking bans during construction to increase capacity.
Cut and cover is only cheaper if you are going shallow. The deeper the tunnel the more expensive it is. If you have to cross an existing subway tunnel you have to go under it - even if we allow you to shutdown the existing subway for months cut and cover the additional depth is more expensive than just tunneling the whole thing. If there is a sewer line across any street you need to reroute that somehow to cut and cover. Same problem for water mains, electric, phone, cable, internet...
It quickly is cheaper to tunnel than to make plans for each thing you need to work around to cut and cover.
