Dutchess County residents were similarly agitated, if less rowdy, at their June 14 meeting about the stadium. Guidance on using the funds issued by the Treasury Department specifically cited stadiums as “generally not reasonably proportional to addressing the negative economic impacts of the pandemic.”

So why, those in attendance 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, according to the Biden administration’s guidance, stimulus funds could indeed go toward investing in things like the stadium.

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

In a recent interview, Mr. Molinaro said that because the funds were one-time money, the county needed to be careful not to create expenses that could not be paid for once the federal funds ran out.

He added that investing in the stadium would produce an ongoing revenue stream for Dutchess County — money that he said would allow the government to pay for the types of programs that Democrats wanted.

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

That explanation has not mollified everyone.

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