Actually the bill was from a Democrat controlled House but a Republican controlled Senate and Regan had no choice but to sign it. It was another one of those, congress can not agree on a budget so we will lump it all together in an ominous spending bill.
You are correct.
Both the left and right side of the “isle” agreed to the Omnibus Budget Reconciliation Act of 1981 that turned funding of mental health back to the states.
Despite that it was a bipartisan effort, liberal coloring books do like to blame that pesky Reagan fellow for anything dealing with mental health from homelessness to mass shootings.