That is not a problem of population density, that problem is inadequate peak capacity in transportation infrastructure. For example if you stack subway lines one under the other simalar to those described by Elon Musk, or increase the capacity of the existing public transport system through through faster & more frequent trains, you can resolve such issues. You just need to commit to spending on the infrastructure to support the population density.
Interestingly, it is policies like prop 13 in California that restrict the revenue needed to invest in the infrastructure to support higher population densities.
I've been to Tokyo several times and always use public transit, even during rush hour and it honestly has never been worse than the light rail in SF. It gets crowded at times sure and I'm sure there are specific times on some lines when it's this bad that you get staff posing passengers in, however it's far away from being the norm. It's usually pleasant and if take it over my commute on a car any day.
Interestingly, it is policies like prop 13 in California that restrict the revenue needed to invest in the infrastructure to support higher population densities.