A couple of points:

Some of these also have non-causal formulations (e.g. principles of least action), but there is always a causal description

Sort of true, although to make a "causal description" of GR one has to do unspeakable violence to the Einstein equation, which simply states that curvature = energy-momentum density. It also excludes many of the popular solutions with closed timelike curves and other anomalies. In any case, if you don't need a causal formulation, or if you can derive it from a non-causal one, then asserting that causality is essential in physics would be reaching.

If physicists don't explicitly talk about causes, it's because the concept is too basic and agreed-on to need talking about.

That's not true. Physicists do explicitly talk about causality, as in, how much of the future can be influenced by the past. Scott Aaronson recently wrote a paper about it.

All that said, however, I believe that what SC and especially WLC meant by causes in their debate was "external causes", more in a sense of a creator, or at least fire in the equations, not the mundane idea of equations of physics being castable in a hyperbolic form. And that vague notion of external causes is what SC objected too.