The MTA is a public authority controlled by the governor. Control was kept away from the city due to the city’s fiscal and corruption problems and the machinations of Robert Moses. The subway and busses were ultimately bailed out by the bridges and tunnels, which print money.
Public authorities are quasi-government entities controlled by the the bond covenants. You have to line-up the interests of the bond holders with whomever controls the authority and is desiring change.
With something as broad in scope and rich as MTA is such a deep well of political capital, it's incredibly unlikely that anyone would give up any control. It's such a large enterprise there probably isn't one roomful of people who actually understand how the organization works. The current situation was created when the whole NYC transit system was completely insolvent back in the bad old days!
Read "The Power Broker" by Robert Caro. It will open your eyes to why things are the way they are.
In New York, thars not the problem. The problem is that a combination of different election laws make it basically impossible to win a primary without the backing of the party leadership itself.
So, primaries in New York are meaningless; there's basically no way to unseat a candidate who's supported by the party, no matter how much they're despised.
^ Bloomberg flipped around parties for convenience of elections. In regards to this topic he would be on the conservative side given that he was constantly butting heads with the transit unions.
Why can't people vote for the candidate of a different party? It doesn't have to be Republican. Why isn't there a "Fix the Subway" single-issue party running candidates that people can vote for?
The internal machinations of political party membership are only relevant if voters no longer have a de-facto choice on election day.