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.
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.