Almost every Subway supports transparent upgrades during operation. In the absolute worst case scenarios you shut down single stations at night. By design almost every Subway is redundant - you at least have to have two lines, one for station bypass and one for returning trains. When you work on any Subway, you shut down one of the lines and manage traffic through the other.
The NY Subway is just the most incompetent, most mismanaged, most technically indebted subway on Earth.
There is no "problem with mass transit" as much as it is a problem with centralized administration of complex things, period. Anything of large scale that impacts many lives but has limited controllers to manage it is both rife for ineptitude / corruption / incompetence and easily ignored on the consumer side because of the complexity.
The same issue turns up all over infrastructure - outdated bridges, power lines, internet infrastructure, water pipes. Central authorities of incompetent taxpayer dollar rent seekers run them, the people are too ignorant / ill informed about deteriorating conditions to care, legacy technical debt builds up over centuries to be impossibly large.
And you won't escape them. The inverse of your scenario - rather than the car breaking down, the road collapses from age - is still wholly applicable. Car travel and vehicular routes are just as mired in corruption and bureaucracy making them a nightmare as trains. Trains breaking down is as easily solved as cars - there are maintenance stations, you take one train off, you put a new one on. The real problem is always when the tracks go bad - the centralization part. You could have a billion trains but not enough track / switching capacity to use them.
In much of Europe, metro systems will shut down for maintenance for appropriate periods. This is massively cheaper than working around a live railway (especially underground), where safety would necessitate many more staff required and more interruptions to work -- if it's even permitted in the particular location.
The period depends on the city. Here in Copenhagen, where the metro runs all night, they tend to shut part of a line from 00-05h on a Monday-Thursday night, and put on buses instead. Cities with bigger metro systems might not need the buses: this was the case the last time I visited Berlin, and a central section was closed overnight.
London sometimes closes parts of lines at weekends, and every 2-3 years or so there is major work on a line (closed entirely) on ~25-30th December.
(To be clear then: the "absolute worse case" is closing an entire line for around a week.)
The NY Subway is just the most incompetent, most mismanaged, most technically indebted subway on Earth.
There is no "problem with mass transit" as much as it is a problem with centralized administration of complex things, period. Anything of large scale that impacts many lives but has limited controllers to manage it is both rife for ineptitude / corruption / incompetence and easily ignored on the consumer side because of the complexity.
The same issue turns up all over infrastructure - outdated bridges, power lines, internet infrastructure, water pipes. Central authorities of incompetent taxpayer dollar rent seekers run them, the people are too ignorant / ill informed about deteriorating conditions to care, legacy technical debt builds up over centuries to be impossibly large.
And you won't escape them. The inverse of your scenario - rather than the car breaking down, the road collapses from age - is still wholly applicable. Car travel and vehicular routes are just as mired in corruption and bureaucracy making them a nightmare as trains. Trains breaking down is as easily solved as cars - there are maintenance stations, you take one train off, you put a new one on. The real problem is always when the tracks go bad - the centralization part. You could have a billion trains but not enough track / switching capacity to use them.