The veto was not always a key tool of executive power; early presidents used it with restraint or, as in the cases of John Adams, Thomas Jefferson, and John Quincy Adams, not at all. Things began to change once Andrew Jackson arrived in office and set about expanding the powers of the presidency. Prior to Jackson, according to UCLA political scientist Scott James, "[s]ettled practice authorized vetoes where Congress's authority to legislate in a given area was constitutionally murky, or where hasty consideration produced legislation with remediable technical deficiencies."Things I didn't know. In any case, Levinson's complaints about against the veto mostly boil down to the fact that, thanks to the electoral college, presidents aren't really very representative of the nation as a whole—they can lose the popular vote, after all, and are often elected by pandering to the preferences of a few key "swing" states—and hence, shouldn't be able to thwart the will of Congress, which is more democratic. That sounds more like an argument for scrapping the electoral college to me, but what do I know.
The veto did not operate "to substitute the president's judgment for the Congress's on national policy matters." In the ensuing decades, however, presidents slowly stopped thinking of themselves as mere enforcers of laws passed by an autonomous Congress. The veto gave them power to block legislation they disliked, and the threat of the veto gave them growing influence over how laws took shape. Soon, presidents were key participants in the legislative process.