It was more than attempted. Wallace was taken out of the campaign, so it was successful to that extent.
His purpose was to "prove his manhood", not to take Wallace out of the campaign.
Had Wallace not been shot, many of those voters, later termed Reagan Democrats, may well have voted for him. There was also discontent with Nixon on the right wing of the Republican Party due to his centrist and even liberal policies: wage and price controls, recognition of Red China, establishment of OSHA and the EPA, promotion of afifrmative action, etc. Ohio Congressman John Ashcroft even staged a quixotic challenge to Nixon in the Republican primaries. Some of the disaffected conservatives might have defected to Wallace as well: by 1972, unlike in 1968, Nixon had established a track record in the Oval Office. Had McGovern won as the result of an independent run by Wallace or Wallace somehow actually won the election, the history of the 1970s and beyond would have been far different.
Needless to say, Arthur Bremer's shooting of the Alabama governor changed the course of American history.