In OTL the response of the Southern (at that time all white) electorate and the treatment of whiet unionists as well as former slaves outraged not only radicals but essentially all republicans
That was why Congress acted like it did
In otl Lincoln though no radical was a little a head of the mainstream.
He may well have acted as the radical congress did, but more effectively because exec and legislative power would clearly be togehter
Together to what effect though? I've already said that Lincoln would sign the Civil Rights and Freedmens Bureau Bills, rather than vetoing them as Johnson did. However, since both were later passed over Johnson's veto, even that only brings them into effect slightly sooner.
OTL, Radical Reconstruction got started in March 1867. Since the Black Codes were only enacted in late 1865, and the massacre in New Orleans was early 1866, Lincoln's support only starts it about a year earlier - hardly enough to make any dramatic difference to the curse of events.
I take your point about Lincoln being in the middle of the Republican Party. Notice, though, that Republicans soon lost interest in getting tough with the South. The political disabilities imposed by Sec 3 of the 14th Amendment could be lifted only by a two-thirds vote in both houses of Congress, yet by 1872, though both Houses were still solidly Republican, these disqualifications
were lifted from all but a handful of ex-Rebs. Mainstream Republicans continued to pay lip service to the Freedmen's rights, but at the end of the day, reconciliation (ie between the
white communities on both sides) was seen as a higher priority. And there is nothing in Lincoln's (thoroughly mainstream) record to suggest that he would not have shared this view. In any case, he will only be in the White House till 1869, after which he is merely a spectator, though a popular one.
Could I recommend William C Harris,
Lincoln's Last Months and
With Charity For all; Lincoln and the Restoration of the Union for some good discusssion of his attitudes at the end of the war?