Well, he didn’t think he did it wrong. Looking back on things and announcing fault of those at the time is a tricky business. Race relations in the US was a complex business, as it was (and is) in many countries (ask any Dalit, for instance.)
At the time:
There were race-riots in Mobile (May 1943), Detroit (February 1942), Beaumont (June 1943), Detroit and Los Angeles (June, 1943) and Harlem (August, 1943). There were explicit (“Jim Crow” laws) and implicit (everywhere not explicit, basically) segregation all over the US – right up until the 1960s.
Many, but by no means all, whites in this country were dead-set against black people having equal status. The upper eschelons of the military were among those. It wasn’t fair, it wasn’t smart, it wasn’t right, but it was fact.
In the military, there were race-riots on Guam (December, 1944), a “mutiny” on Hawaii (July 1944) and at Port Chicago (November, 1943), a SeaBee ‘hunger strike’ (February 1945), a ‘mutiny’ trying to break off-duty segregation - illegal in the military - in Kentucky (April 1945), “The battle of Brisbane” (US African-Americans, Australian whites) (March 1942), another in Townsville, Australia (May 1942), Bamber Bridge, England (June 1943). And those are the ones we know about. Ask any older African-American about the endless small indignities of that era. (And it ain’t “just like today”. It just isn’t.)
The military medical service segregated blood until 1950 (that story about Charles Drew is not true, by the way.)
The British tried to keep African-American soldiers out of the British Isles entirely for racist reasons. (Ask anybody non-white about the Brits.).
The NFL and MLB had ‘unofficial’ (but complete) racial segregation up until 1947 (the NFL 1946).
… now, if you’re the US President, and you’re trying to fight a war - primarily with white soldiers, sailors, airmen and Marines - are you willing to cause the massive upheaval by desegregating units? Are you going to desegregate units in combat? Are you going to replace officers who refuse to integrate their units? If so, will the replacements be inexperienced and get more men killed? Is this going to get a N*-C* war going in units?
How exactly do you do it? How exactly do you do it and tell grieving fathers, mothers, wives, and children that their loved one died not in defeating Germans, Italians, or Japanese - but because some idiot in the same uniform of a different race decided to kill him?
FDR, and Truman after him, played a much more subtle game. He allowed African-Americans to prove, indisputably, that all the scurrilous thinking about them (“they won’t fight”, “they can’t do complex things”, on and on) was wrong, and completely wrong. He allowed (and indeed influenced) African-American units to be put into combat, and in combat zones. He let African-Americans prove themselves; when integration came, it was not a white man’s gift, it was the black man’s reward.
Belief that a mere stroke of a pen changes people’s thinking is just naive. Know a man, know he is a good man, and you can’t hate him.