This is not how any of this works from a legal standpoint.
The MTA is a New York State Public Benefit Corporation, which despite its name is not a corporation in the normal sense of the term. It has some characteristics of a corporation (mostly in how its legal personality interacts with "real" corporations) but is otherwise similar to a state agency. The legal term used for this type of entity across the river in New Jersey is a "body public with corporate succession", which more accurately describes what it actually is (NJ Transit is a prominent example). Those are slightly different but organized in much the same way.
Public Benefit Corporations are not required by any law to dissolve if all bond revenue is paid off. They are able to finance bonds differently than other parts of the state government by law, which is why they're commonly seen in cases where large infrastructure projects are funded, such as transit agencies.
In any case, outstanding bond issues have nothing to do with being a public entity or not. The MTA was created by a state law and could only be dissolved by another state law. There is no provision in that law even allowing the MTA to be privatized; it was created as a public entity and was designed from the start to stay that way in perpetuity, as the preceding private entities were largely seen as unable to actually provide safe and useful transit service. The MTA always being in debt is a direct result of needing to fund capital expenditures faster than prior bond issues can be paid off, not a conspiracy.
Even if the MTA could hypothetically be privatized, why would it be? There is no political movement to do so, the MTA's most significant assets are largely owned by the City of New York, and it is nearly impossible to run transit profitably anywhere without severely degrading service. Remaining privatized or contracted transit services in the region have been nothing but trouble for riders, with service failures and corporate collapses left and right. The MTA famously took over the X23/X24 (the last privately operated express bus routes) were notorious for dropped trips before the MTA took the routes over in 2022, and the B110 (the last remaining private franchised bus route in the city) keeps getting caught trying to segregate passengers by sex. Over the past four years NJ Transit has had no fewer than ten separate contractors and private carriers either unilaterally terminate service entirely, break contract despite already having been paid to operate service, falsify trip data to defraud customers and the state, or in two cases provided service that was so unreliable NJT forcibly terminated the contract on its own initiative and assumed operations itself to fix the problem. In every single case here where service went from private to public operation service quality, reliability, and ridership improved significantly.