The moment you introduce a method by which teachers can be ranked in simple order and in the short term, I can guarantee you, principals and other admins are going to get swamped by parents who will be trying to jockey to get their kids into the classrooms of the "most effective" teachers. We're talking stampede.
This will also undoubtedly fan the flames for those arguing for merit-based hiring/firing and/or pay decisions. The district (or better said, the union) has been the subject of several unflattering news stories that point out how incredibly difficult it seems to be to get rid of unwanted teachers. Now that parents and politicians will have a chart to point to, there's going to be even more calls for the district and union to dump the chaff and rethink its hiring and training procedures. Again.
Couple of things just to get out of the way. First, the district takes a lot of hits for being unwieldy, mismanaged, bloated, inconsistent, etc. and all of that is probably true. The power plays between the mayor's office, the district board, and other civic leaders is like a French period drama without all the wigs. All this is happening against a backdrop of nearly 700,000 students and over 80,000 employees, in a post-Prop 13 budgetary environment. In short, there's no easy answers here at all. I'm not saying the district deserves a pass but let's just say that running the LAUSD efficiently and effectively probably isn't as easy as running Bell into the ground.
Second of all, before some fruit fly in the comments tries to argue that the pink elephant in the room is illegal immigration, I think that's a whole 'nother conversation, and one that requires more discussion than just indiscriminate scapegoating that fails to take into account A) budget cuts to public school education over the last few decades in CA, especially after Prop 13 rejiggered property tax (a heavy base for school funding) and the ability to raise taxes to help fund education (and everything else in the state), B) the social mission of public education for all students, regardless of their (or their parents') immigration status, and C) how much of the district's woes are due solely to the size of its student population. It may be a conversation worth having but it's separate from this issue.
Back on topic: to me, the real issue here is a core problem that exists throughout the educational system, not just K-12 but certainly in college too:
Should teachers be evaluated for their teaching? Yes.
In other words, I think almost anyone would agree that teachers—especially tasked with such an important job as they have—absolutely need to be given feedback and be evaluated for their effectiveness as teachers. I can't imagine anyone suggesting that acquiring a teaching credential means you're suddenly not subject to the same kind of on-the-job scrutiny as any employee, anywhere would be.