(Preface: the Rad Trad is not a universalist)
There has been some resurgence of interest in a video, posted below, by one Fr. Robert Barron about the population of hell. Fr. Barron gives an enthusiastic reaction to the writings of mid-20th century writers who resurrected interest in Origen's "restoration" (apokastasis) view, that all creation, including all persons, will be restored to a proper relationship with God.
Origen and Fr. Barron clearly fall opposite the view of the consensus of Western Doctors such as Ss. Augustine and Thomas Aquinas, whose view of Original Sin began with the thesis that all the world is damned, but God, in His kindness, plucks a few persons out of the eternal stir fry. St. Leonard of Port Maurice recounts a tale in his famous sermon on the subject of a man who died at the same moment as 30,000 other people, five of which were saved (0.017%); the Rad Trad personally finds this number absurd given how high infant mortality rates were prior to the 20th century. The Council of Florence (Cantate Domino) and Boniface VIII (Unam Sanctam) certainly attempted to limit the number saved to a minute figure.
The Eastern Fathers have a more varied view on the matter. St. John Chrysostom certainly thought a very small number would be saved ("Out of this thickly populated city with its thousands of inhabitants not one hundred people will be saved. I even doubt whether there will be as many as that!"). Ss. Isaac of Syria and Gregory of Nyssa thought all would be saved. Does it have to be all or nothing?
Given the presence of a convent of Feeneyites in the Rad Trad's area, the Rad Trad actually finds himself defending universalism as a hypothesis (as opposed to a "reasonable hope"). Saints in the earlier days of the faith found no trouble reconciling universal salvation with Christ's words on Hell and the damned. Sometimes the meaning of things Our Lord said becomes obscure in translation. For instance a Greek priest once told the Rad Trad the "The King of Heaven is upon you" has a stronger meaning in the original, almost saying "The Kingdom of Heaven is immanently accessible to you here on earth." The Rad Trad, not a believer in universal salvation himself, does think that, if constructed along orthodox lines, the salvation of all can be a reasonable idea. Promoting it to a hope—an expectation for God—asks too much.
What frustrates the Rad Trad when he does occasionally defend the viability of this thesis is that those who advance it today do not do so on orthodox grounds. Origen was probably condemned unjustly at the Fifth Ecumenical Council three centuries after he died. He was the first "theologian" in the modern sense of the word and was bound to err on many matters, given that he was treading new territory. He did not have the opportunity to hear the Church adjudicate concerning his novel doctrines and adjust them accordingly. His restoration theory, which informed his views on salvation, were what brought about his condemnation centuries later and yet that is precisely what interests Fr. Barron and others in the modern day. The presentation of this view is supported by vague notions of "mercy" as the expense of justice, almost as to say that God is a giant marshmallow. This contrasts sharply with St. Gregory of Nyssa's view that people will be saved because the soul belongs to God, but that many souls will have to undergo a painful purification from their sins and the passions of the world to be worthy of Heaven. In this sense Gregory's views differ from those of Origen, although the Orthodox are not very wild about either one. Whereas Gregory's view maintains God's mercy and justice, the more modern view, rooted in Origen, overlooks or eschews justice in favor of mercy alone.
Is everyone saved? Is nearly no one saved? The Rad Trad thinks the honest, humble answer is "We don't know who or how many are saved." And given that we do not know, we Catholics would do well to take special care for our own souls and an interest in the souls of others. That is the evangelical call.