The assassins were insane...
The OTL's successful assassins in the 60s were insane and the OTL's failed assassins of the 70s and 80s were insane, but that doesn't mean every potential assassin in an ATL is insane.
When you look at presidential assassins, either successful or failed, the majority were mentally unbalanced. There are significant exceptions to that however.
Booth's murder of Lincoln was a political act with defined goals and part of a larger conspiracy. A conspiracy of fools certainly, but not a conspiracy of the insane. When he murdered McKinley, Czolgosz saw himself as part of the worldwide anarchist movement. Like the other anarchists who had been knocking off presidents, monarchs, and other officials for decades, Czolgosz was making a political statement. After he took a few shots at FDR, Zangara made noises about killing all capitalists but it's generally believed he had been paid by criminal elements in Chicago to kill Mayor Cermak, something he managed to achieve while somehow also "missing" FDR. The attempt on Truman was carried out by Collazo and Torresola, both activists for Puerto Rican independence, and was framed by them through out their trials as a deliberate political act.
You can see that not every assassination or assassination attempt on a US president has been the work of madmen for reasons only the insane would understand. Occasionally, those acts have been for political purposes.
... and the public believing it to be a liberal conspiracy is ASB.
The public believing that 1970s OTL failed assassins Fromme or Moore were part of a liberal conspiracy is completely ASB.
Those women are not the only potential assassins in an ATL, however, and other those other assassins can very well have motives which don't arise from psychiatric issues.