I actually don't quite like either side's position, although it shouldn't surprise much of anyone that I'm more sympathetic to the liberal argument than the conservative one.
So I'll start with my problem with my own side. Essentially, the notion that the admissions exam should play no role at all does seem a bit suspect to me. If nothing else, I don't think it does anyone much good to risk bringing in new students who are not prepared for the challenge facing them, nor do I think it does the school much good to give them a challenge their teachers may not be as ready to face as they think they are. Unfortunately, I don't think it's possible to keep the test as a means of ensuring that all of the prospective students are ready to succeed without it also being used to favor those prospective students that already have more resources and time to spend on preparing for it, but I also think that having something is necessary.
That said, the notions on the other side that teachers' unions are responsible for the inadequacy of our existing public schools - or that the solution is to abandon them entirely and rely on charter schools and the free market - is not even remotely true. Part of the reason I talked above about handing teachers a challenge they may not be ready for is that it turns out a lot of the really good schools that get cited by conservatives as examples of the free market's success got that way by not having to deal with students that might bring their numbers down. If the goal is to help everyone succeed, we don't have that option when dealing with the entire system.
In that sense, I think the point that the one NYT editorial makes is closest to the right answer - we need to be looking to make all of our schools into elite schools, and if that costs us a lot of money and effort, then so be it.