No there is no flagship.
When the SUNY system was developed, long after the flagships of most states were well established and were attracting NY students, the officials promised that the SUNY system would not compete with private schools for students. Promising the no-compete was one of the ways that SUNY system was able to counter the resistance to establishing the system. The officials promised that the schools would be designed expressly for students who could not afford private schools and who would otherwise not have access to higher education.
The SUNY officials promised they would not target students who could attend private schools. The utilitarian look to the buildings and the general aesthetics of the campuses are probably attributable to that promise. It is ironic that the PR now seems to be geared towards trying to convince the public that SUNYs are the same as private schools, just cheaper. So do the SUNY schools compete with private ones or have they stayed true to their word?