Unless the monarch is a landed oligarch in their own right. See France to see this at its logical conclusion.
I think this is where different kinds of monarchies come into play. Compare France to other countries, and you'll see how different monarchies can really be.
Though, in general, whether a monarch stabilizes a country depends on how a monarch came in power. The Second Mexican Empire, for instance, rose because of a French invasion - this immediately makes it unpopular. Same goes for the Bourbon Restoration - as it was installed by a coalition of monarchies (and revoked many revolutionary principles), it was unpopular. But if the monarchy is seen in other ways, and rises differently, such as with the First French Empire, it can indeed be a stabilizing force.
However, I do think the stabilizing effects of monarchy are quite overrated, if only because things would be more "different" with more monarchies, and so people like to include them everywhere.