In the Washington Post piece, the case for this conclusion is patchy. For example, only one state's secession document is cited explicitly (the Texas document). It's quite possible that the case is much stronger and more comprehensive in Loewen's book, The Confederate and Neo-Confederate Reader. After all, one can only include so much in a short essay.
Despite the patchiness of the essay, at its heart is an interesting line of argument, one that's nicely summarized in the following excerpt:
In its “Declaration of the Causes Which Impel the State of Texas to Secede From the Federal Union,” for example, the secession convention of Texas listed the states that had offended the delegates: “Maine, Vermont, New Hampshire, Connecticut, Rhode Island, Massachusetts, New York, Pennsylvania, Ohio, Wisconsin, Michigan and Iowa.” Governments there had exercised states’ rights by passing laws that interfered with the federal government’s attempts to enforce the Fugitive Slave Act. Some no longer let slave owners “transit” across their territory with slaves. “States’ rights” were what Texas was seceding against.The point here is that, by opposing the right of these states to pass laws impeding enforcement of the Fugitive Slave Act, Texas was thereby standing against the right of states to make their own laws on these matters. Texas was upset because these states weren't capitulating to the Federal Government, but were instead standing up for what they believed in the face of contrary Federal mandates. In short, Texas seceded in part because they were upset about Northern states that expressed too much autonomy in the face of a Federal mandate that served the institution of slavery.
This point does not, of course, entail that Southern States didn't invoke state's rights. What it entails is that to the extent that they did invoke state's rights, it was a selective invocation. If they invoked state's rights to defend their right to keep slaves, but opposed state's rights whenever a state made laws and policies unfriendly to slavery, what we have is evidence that the real issue was not a principled defense of state's rights but a consistent defense of slavery.
None of this means that slavery was the only issue motivating Southern secession. What it does mean is that slavery was a deeper and more central issue than state's rights. Were state's rights more important to Southern states than preserving the institution of slavery, wouldn't they have affirmed the right of states to refuse to facilitate the return of runaway slaves? That, on the contrary, they were denounced states for doing this suggests they cared more about preserving slavery against any forces that threatened it--including both federal and state-initiated threats--than they cared about state's rights.