I have no objection to a Constitution-based veto, even if it is unclear whether the president should, in such circumstances, be allowed the last word. Rather, I am concerned about a system--like the one the United States currently has--under which the president becomes, in effect, a third legislative chamber, able, with the stroke of a pen, to negate the views of up to 289 House members and 66 senators.
What is so wrong with permitting the White House to act as another legislative branch? Put simply, the veto tilts the balance of power in Washington too far toward the status quo. It is already difficult enough to gain the consent of both the House and Senate on legislation, and the veto exacerbates this problem. Advocates of change must capture the House, the Senate, and the White House, whereas defenders of the status quo need to win over only one. This sets a high barrier to reform; and paralysis on key issues--such as health care and immigration--is too often the result.
If we repealed the veto, it would move the US toward a quasi-parliamentary system -- an action that would have far-reaching consequences for the organization of parties and executive power in this country. But it would not necessarily prevent legislative "paralysis."
Set aside the president for a moment and start with the Senate, which is also anti-majoritarian. As Keith Krehbiel's book Pivotal Politics points out, the approval of any policy in the Senate requires that the pivotal voter in a filibuster prefers that policy to the status quo. If the proposal is to move policy in a conservative direction, then the 41st most liberal senator is the filibuster pivot (and vice versa for moving policy in a liberal direction).
The veto comes in when Congress passes a proposal that the president does not prefer relative to the status quo. If Democrats took back Congress, for instance, then the relevant pivots in the next Congress would be the 67th most liberal senator and the corresponding House member, who would both have to prefer the policy to the status quo to override the president's veto. If the president did not have veto power, the relevant pivot would be the 60th most liberal senator (and the median House member).
This shift would be the most basic legislative consequence of the elimination of the veto. Its magnitude would vary depending on the ideologies of the relevant members of Congress -- it could be quite substantial or relatively modest. According to the DW-Nominate estimates of legislative ideal points for the 108th Congress, for instance, the 60th most liberal senator was Richard Lugar of Indiana and the 67th most liberal was Jim Talent of Missouri. In the House, the median voter was Jack Quinn of New York and the veto pivot would have been Kay Granger of Texas.
It's unclear to me why Levinson isn't targeting the filibuster instead. Abolishing it (while unlikely) would actually be easier than eliminating the veto since the filibuster is not constitutionally mandated, and doing so would have a similar -- or perhaps greater -- effect on the range of legislation that could be passed into law.
(Note: All of Krehbiel's results assume a unidimensional liberal-conservative policy space. See Charles Cameron's Veto Bargaining for much more on the dynamics of vetoes and veto bargaining. And of course don't forget that parties can pressure pivots into voting with them on key legislative issues.)