FDR tried to implement mandatory medical insurance for the needy, funded by a payroll tax. Not "universal healthcare." (I was incorrect on that above, because I was taking you on your word.) Truman wanted the same thing, but for everyone; this was decried as "socialist" (because we hate the Commies). Private insurance, doctors, hospitals all lobbied hard against it, as I'm sure they still do.
There's also a nice chart here which shows who controlled the Presidency and each house of Congress since 1901. Under FDR, Democratic control handled the Great Depression and WWII. JFK dealt with the Cuban Missile Crisis and Johnson did civil rights and Medicare/Medicaid. Obama did the Affordable Care Act.
Since you're hyper-focused on universal healthcare, can you name any Republican expansion of access to healthcare in the US? Ever?