Seattle has dug tunnels trying to help congestion, but it hasn't helped much. The first tunnel was putting buses under the street in downtown. Now they are in the process of replacing an entire freeway with a tunnel.
The Seattle metro area is a hodgepodge of transportation experiments, many of which don't help and some make things worse. I-5 is 6 lanes south of downtown and goes down to 2 lanes in downtown with a set of reversible express lanes that only amount to 2 or 3 lanes. The transit engineers in the 1990s were still thinking about how to get people into Seattle in the mornings and out to the suburbs as efficiently as possible when just as many people were living in Seattle and working in the suburbs as were coming into the city. Microsoft and the cellular boom moved most of the high tech out to the suburbs and the hipsters working in high tech wanted to live in Seattle. There are also a large number of people who commute from one suburb to another and never go near downtown Seattle in their commute, but the whole transit system is stuck in the 50s when most people worked in Seattle and lived in the suburbs.
We moved to the Portland area in large part because of the traffic in the Seattle area which was just getting worse.
Seattle put buses in its first transportation tunnel, but I think Elon's idea could tie in well with the vision of autonomous EV taxis. If the tunnels were limited to EVs, the need for ventilation would be greatly reduced. You still need some fresh air cycled through, but much less if there are no fossil fuel vehicles. Seattle's bus tunnel uses hybrid buses that only run electric on the underground part of their routes. t appears light rail runs down there now too.
People on the surface tend to resist tunneling projects and it can cause problems when the tunnel is under construction. When Seattle was building the bus tunnel, there were small tremors in downtown fairly frequently and one day they accidentally tunneled into an underground river nobody knew was there. The flood got into the power lines for downtown (which are also underground) and it knocked out power to the entire downtown core for close to a week.
The problem with congestion may start to take care of itself as autonomous cars and trucks begin to take hold. With autonomous trucks the bulk of truck traffic in cities can be in the middle of the night which will cut down on daytime congestion. In the 1984 Olympics they required all truck traffic to be at night and it really helped. Additionally with lots of essentially autonomous taxis running around, overall traffic congestion may get much better. Autonomous vehicles that are able to communicate with one another would also be able to run much closer together and at much higher speeds.
It would be a race to see which came first, the bulk of the impact on traffic from autonomous cars, or the permits for the first tunnel boring project. With environmental studies and everything else involved, it would take at least a decade to break ground on one of Elon's tunnel boring projects. If traffic congestion in large cities had begun to decline by then, city planners may opt not to do it.