But I'm not sure how to reconcile that with modern "rightists" who want to burn down the system and aren't conservative in the lowercase-C sense.

The Republican party (and conservatism as a movement) are full-blown reactionaries. I like this passage from Corey Robin'sThe Reactionary Mind:

People who aren’t conservative often fail to realize this, but conservatism really does speak to and for people who have lost something. It may be a landed estate or the privileges of white skin, the unquestioned authority of a husband or the untrammeled rights of a factory owner. The loss may be as material as money or as ethereal as a sense of standing. It may be a loss of something that was never legitimately owned in the first place; it may, when compared with what the conservative retains, be small. Even so, it is a loss, and nothing is ever so cherished as that which we no longer possess. It used to be one of the great virtues of the left that it alone understood the often zero- sum nature of politics, where the gains of one class necessarily entail the losses of another. But as that sense of conflict diminishes on the left, it has fallen to the right to remind voters that there really are losers in politics and that it is they— and only they— who speak for them. “All conservatism begins with loss,” Andrew Sullivan rightly notes, which makes conservatism not the Party of Order, as Mill and others have claimed, but the party of the loser.

The chief aim of the loser is not— and indeed cannot be— preservation or protection. It is recovery and restoration.

And from another section:

There’s a fairly simple reason for the embrace of radicalism on the right, and it has to do with the reactionary imperative that lies at the core of conservative doctrine. The conservative not only opposes the left; he also believes that the left has been in the driver’s seat since, depending on who’s counting, the French Revolution or the Reformation. If he is to preserve what he values, the conservative must declare war against the culture as it is.