  In this article, we apply slope detection techniques to study properties of
toroidal $3$-manifolds obtained by performing Dehn surgeries on satellite knots
in the context of the $L$-space conjecture. We show that if $K$ is an $L$-space
knot or admits an irreducible rational surgery with non-left-orderable
fundamental group, then the JSJ graph of its exterior is a rooted interval.
Consequently, any rational surgery on a composite knot has a left-orderable
fundamental group. This is the left-orderable counterpart of Krcatovich's
result on the primeness of $L$-space knots, which we reprove using our methods.
Analogous results on the existence of co-orientable taut foliations are proved
when the knot has a fibred companion. Our results suggest a new approach to
establishing the counterpart of Krcatovich's result for surgeries with
co-orientable taut foliations, on which partial results have been achieved by
Delman and Roberts. Finally, we prove results on left-orderable $p/q$-surgeries
on knots with $p$ small.
