Dutchess County residents were also restless, though less rowdy, at their June 14 meeting about the stadium. Guidelines on the use of funds issued by the Treasury Department specifically cite stadiums as “generally not reasonably proportionate to address the negative economic impacts of the pandemic.”

So why, those present asked, was this happening?

Marc Molinaro, the county executive, defended the spending, saying Dutchess County had identified $33 million in lost revenue as a result of the pandemic and that, under Biden administration guidelines, the funds stimulus could indeed be used to invest in things like the stadium.

“It’s basically any structure, facility, thing that you own as a government that you can invest those dollars in with great latitude,” Molinaro said.

In a recent interview, Mr. Molinaro said that because the funds are one-time money, the county must be careful not to create expenses that cannot be paid once federal funds run out.

He added that investing in the stadium would produce an ongoing source of revenue for Dutchess County — money he says would allow the government to pay for the kinds of programs Democrats want.

The investment, he said, “allows us to create 25 years of revenue that we can invest in expanding mental health, homelessness and addictions services.”

This explanation did not appease everyone.

“I was just devastated that we spent the money this way,” Ms. Kearney, the Democratic lawmaker, said in an interview. “It was such a betrayal of our community. So inappropriate and grossly deaf to the needs of the people of Dutchess who suffered.