Can Agnostic Liberal Democracy "Work"?

We'll have to join Patrick Deneen and Abp. Vigano in answering that question:  nope.  (For that matter, Meaning in History asks the same question framed in and largely limited to "Trump/No Trump" terms here.  And he's not an optimist, either.)

+Vigano begins with a searing critique of today's Catholic Church (and by extension, the rest of Establishment Christendom).

The doctrine of the kingship of Christ constitutes a discrimen between the Catholic Church and the “conciliar Church;” indeed, it is the point of separation between Catholic orthodoxy and neomodernist heterodoxy, because the followers of laicism and liberal secularism cannot accept that the Lordship of Our Lord extends to the civil sphere, thereby removing it from being subject to the arbitrariness of the powerful or the will of the manipulable populace. ...

Short and to the point, eh?

He expands on it.

...Yet the very idea that authority has its foundation in a transcendent principle was not born with Christianity, but is part of our Greco-Roman heritage. The same Greek word ἱεραρχία [hierachy] indicates on the one hand the “administration of sacred things,” but on the other it also refers to the “sacred power” of authority, where the commitments connected with it significantly constitute a λειτουργία [liturgy], a public office of which the State takes charge. 

Similarly, the negation of this principle is the prerogative of heretical thought and Masonic ideology. The laicity of the State constitutes the main claim of the French Revolution,¹ for which Protestantism provided the theological foundations, which then changed into a philosophical error with the advent of liberalism and atheistic materialism. 

This vision of an entirely coherent and harmonious whole that spans the passage of time and crosses the boundaries of space, leading humanity to the fullness of Christ’s Revelation, was proper to that Civilization whose removal and cancellation is desired in the name of a dystopia that is inhuman because it is intrinsically impious, since it originated from the inextinguishable hatred of the Adversary, eternally deprived of the supreme Good because of pride and rebellion against the Will of God. ..

You can read the entire essay at the link.  But that's really all you need to know about why liberal democracy cannot work for long.

Can we hope for the solution which would be to re-install at least the understanding that Christ is King of time AND eternity?

Hmmmmm.

Choose well, friends.