You can’t really have it both ways; that teachers are the critical linchipin of successful schools and should be paid accordingly AND they aren’t responsible at all if students aren’t achieving.Sure you can; it's a phenomenon known as 'necessary but not sufficient conditions.'
For example, think about what a human needs in order to stay alive: food, water and air. Each of these is necessary. Without it we would die. But none of them alone is sufficient. We need all three to live.
Now imagine the quote, rewritten: ""You can’t really have it both ways; that food is the critical linchipin of survival and should be paid for accordingly AND that it isn't responsible at all if people aren't surviving."
In fact, if you have the best food in the world, and no air, you will not survive, and it is not at all the fault of the food.
The same is the case with teaching.
I've seen study after study that shows that the main predictor of educational achievement is socio-economic status. Children of richer parents do better in school.
We could speculate about why this is the case. I've seen some people argue that richer parents have greater expectations. Others argue that richer parents provide access to books and computers. Still others say that children of richer parents have more free time (they don't have to work) and have fewer disruptions in their home lives.
That said, it does not follow that teachers are irrelevant to their learning. Children of rich parents, if they are not taught, are much less likely to learn. As things are right now, a teacher is the major conduit of learning for a child, no matter what their socio-economic status. This is why teachers are (or ought to be) provided to all children.
What could be said here is that factors both outside the school and inside the school are necessary for aducational achievement. Take one or the other away and learning does not happen. But neither, by itself, is sufficient. A student needs both.
Hence, it can be true to say that teachers are important to shaping student achievement. Without them, students will not achieve. But it can also be true to say that teachers are not ultimately responsible for outcomes. Other factors are essential as well, and teachers can do nothing about them. A teacher cannot, for example, ensure that his or her students have rich parents.