Well, the president does not introduce legislation either. It is up to the presidential and gubernatorial leaders in Congress to introduce the President's and the governor's bills. I don't think the term "introduce legislation" means a lot from a practical standpoint, but I don't know how other states fare in that regard. If the governor does not support a bill, it cannot pass without 2/3 plus of both houses on an override. And no gubernatorial veto has been overridden since 1979.
My memory of Political Science class, which happened around the time of the invention of dirt, is that the ability to introduce legislation is something that most governor's have the state constitutional power to do. Only a handful of governors don't have that power.
It works okay as long as the governor has a friendly Speaker and Lt. Gov. which is why I speculate Bush and Bullock were so close even though they were miles apart politically. I honestly never figured that out completely other than perhaps Bullock was, above all, pragmatic and I think he saw the political shift even before it happened. Bullock was one smart pol.
But you're absolutely right, no veto has been overridden in a long time, but then a lot of what the gov wants has never, ever seen the light of day or gotten more than five minutes in committee. Witness Bush's proposed tax increase. DOA.