Many of the world's oldest subways have solved this problem through the use of specialized technology transition teams whose sole job is to develop minimal downtime solutions to upgrading technology. I'm no expert on these technical teams...I merely know of their existence through my own personal obsessive curiosity about infrastructure. But at least in terms of having relatively modern performance and technology, Paris, Madrid, Hamburg, Tokyo, Berlin, Osaka, Barcelona, and even Moscow don't seem to have any problems. Some might run what appears to be really old equipment...but it runs on time without any significant downtime. And with the exception of Tokyo (which has severe overcrowding problems which exceed the natural limits of subways) and London (which has extreme geographic, geological, archeological, electrical, and real estate constraints), they have mostly never let these problems impact the daily service.
New York is probably the most easily upgraded of these legacy subway systems, due to the insistence of the original subway companies on triple tracking. Not only does the triple track allow for express trains, but it also allows for zero downtime maintenance during non-peak hours. The electrical systems aren't a barrier like those in some cities. Even Tokyo and London, which have worse constraints, have fared better. If technology transition teams exist, I haven't heard of them. Upgrade engineering appears to be outsourced. The costs that the MTA quotes for transition seem absurd. Once again, I'm no expert on the engineering involved, but the upgrade costs are approaching new construction costs in other countries.
We know we have a major cost problem in US transit infrastructure construction. But this is not just construction. We also have a problem with maintenance costs, operating costs, capital costs, etc. Everywhere you go in America, we have critical transit infrastructure that is hobbled by costs. Where is the cost management? Why can't we get system-wide cost audits? If annual audits are required of publicly listed companies, why isn't it required of public organizations funded by tax dollars?
I'm sick of the explanation that our problem is funding...its not. Our problem is mismanagement. When you manage your systems well, nobody has any problem sending money your way. The demand curve exists for government funding just as much as it does for markets. When you provide more for a given amount of money, you generally get more money. Give us better management.