The search function is so wonky on this site I'm not sure where to look, but were people this angry about Jesse Green's picks in previous years, or is it because he posted his article before the nominations instead of after? Mentioning shows he thought should have been nominated that weren't eligible for the Tonys has long been part of the NYT's schtick, and predates Green:
https://www.nytimes.com/2016/05/15/theater/tony-award-winner-predictions.html
"Ben Brantley and Charles Isherwood, the theater critics for The New York Times, make their Tony Award choices. For the Should Have Been Nominated category, the critics were allowed to expand the potential nominees to Off Broadway productions, which are ineligible for the Tonys."
https://www.nytimes.com/2018/05/09/theater/tony-awards-ballot-who-will-win.html
"Ben Brantley and Jesse Green, the chief theater critics for The New York Times, make their Tony Award choices. For the Should Have Been Nominated category, they were allowed to expand the potential nominees to productions ineligible for the Tonys, including those Off Broadway, which are designated by an asterisk."
https://www.nytimes.com/2019/05/14/theater/tony-awards-critics-picks.html
Ben Brantley and Jesse Green, the chief theater critics for The New York Times, make their Tony Award choices. For the Should Have Been Nominated category, they were allowed to expand the potential nominees to productions ineligible for the Tonys, including those Off Broadway, which are designated by an asterisk.
https://www.nytimes.com/2022/05/18/theater/tony-awards-2022.html
Which brings us to the Should Have Been Nominated category. A larkish Times tradition allows me to include not just other work seen on Broadway but also (as indicated by an asterisk) artists and productions hailing from Off Broadway and other, odder climes.
https://www.nytimes.com/2023/05/10/theater/tony-awards-2023-broadway-critics-picks.html
3. We admit that the “Should Have Been Nominated” category is weird. It salutes work that was eligible for nomination — but also, indicated by an asterisk, work that wasn’t. That includes non-Broadway productions, as long as they opened, like the legit nominees, after May 1, 2022.
https://www.nytimes.com/2024/05/14/theater/tony-awards-2024-broadway-critics-picks.html
"3. The “Should Have Been Nominated” category obviously includes Broadway work that was eligible but spurned. Less obviously, it also includes work from Off Broadway and beyond (indicated by an *asterisk*) that’s totally ineligible for the Tonys, just because."