It wasn't as close as it looked:
"The closeness of the balloting may in itself be deceiving. Considerable evidence exists that other senators stood ready to vote for acquittal if their votes had been needed. As early as May 18 the Chicago *Tribune* asserted that the President's friends laid claim to four more votes in case of necessity, and the substance of the story was confirmed shortly after the trial by Samuel Randall, the Democratic Congressman from Pennsylvania. On August 3, Johnson himself wrote to Benjamin Truman that [Edwin D.] Morgan [of New York] had been one of the Republicans in question. In 1913, Senator Henderson also asserted that Morgan had been the reputed swing voter. Because of the intense pressure, he voted to convict, but would not have done so had his vote made any difference. Some years earlier the Missouri senator told William A. Dunning that Waitman T.] Willey [of West Viriginia] had also been ready to switch, a point he later reiterated to Trumbull's biographer, Horace White. He also mentioned Sprague as one of the senators willing to change, and John Bigelow learned that [James W.] Nye [of Nevada] had been another. In short, Johnson's victory was assured long before the vote was taken. A sufficient number of moderate Republicans stood ready to acquit him, come what might." Hans L.Trefousse, *Impeachment of a President: Andrew Johnson, the Blacks, and Reconstruction* (Knoxville: Univeristy of Tennessee Press 1975), p. 169.
But suppose there had been enough votes for conviction? It is often argued that it would bring about a great change in the balance of power from the executive to the legislative branch. But it is hard to see whether it would in fact set much of a precedent, given that (1) it is rare for a party opposing the president to have not only a majority of the House but also a two-thirds majority in the Senate--so 1868 is unlikely to be repeated; (2) the Republicans were about to get control of the White House with General Grant, and might quickly forget all their complaints about the dangers of executive usurpation once he entered the White House; and (3) eventually Johnson's impeachment *and conviction* might be just as discredited in the public mind as his impeachment was in OTL. After 1876, it was widely believed (at least by whites), rightly or wrongly, that Radical Reconstruction had been a failure and that Johnson had shown courage and some insight (though certainly not tact!) in resisting it. I see no reason that the same feeling would not have developed had Johnson been convicted.
Of course it is true that Ben Wade would be Acting President for several months, and during those months, theoretically Congress might pass ultra-Radical legislation which Wade would sign--e.g., land redistribution a la Thaddeus Stevens. But I doubt they would do so--they wouldn't want to do anything to endanger the prospects for General Grant's election, especially after the 1867 elections showed widespread Northern opposition to black suffrage. (Maybe the Radicals' real window of opportunity would come in a lame duck session after Grant's election and before his inauguration! But really drastic measures like land redistribution divided even the Radicals; and in any event, even if they were enacted, violence by the Klan and other groups might make the freedman's theoretical right to forty acres and a mule as meaningless as his theoretical right to vote...)