Well, I've found it, and I've also found why it was so hard to find despite the bounty that b_jonas generously offered. (Thank you! I just wish that someone had been able to win that bounty.)
The story is With Flaming Swords, by Cleve Cartmill -- not a well-known author. The story was originally published in Astounding Science Fiction in 1942, but I almost certainly read it in A Treasury of Science Fiction, edited by Groff Conklin (1948). I recall other stories from this anthology, and though I did read and reread it in the 1960s, it must already have been on a library shelf for quite some time. The anthology was reprinted in 1951 and 1980, and the story was reprinted in collections of Cartmill's work in 2003 and 2011, but it was probably never well-known. Cartmill is best known today for Deadline, a 1944 story in which he described the atomic bomb in some detail, to the alarm of the FBI.
The way I found With Flaming Swords again after so many years was by a brute-force method. I had already posted a list of books that I've read on Goodreads, and tagged them in several ways. One of the tags is "anthology", so I began looking through them, and then checked the ISFDB for the titles of short stories contained in each one, and searched on Google for reviews of the stories with those titles. I figured that I would rediscover enough short stories to make this worth my while, and that strategy was working nicely when, son of a gun, there it was in one of the first few anthologies that I looked at, out of more than 200.
Jamie Todd Rubin reviewed With Flaming Swords among several other stories on September 17, 2012:
With Flaming Swords by Cleve Cartmill
Blurb: The accident that a new
weapon killed some and mutated others produced–the Saints, the
self-appointed, selfish, self-adulating Saints, who rules the world
with decidedly unsaintly violence and -- with flaming swords.
I wasn't sure about the identification from this, but Jacques d'Emal (1994, Anticipating the Atom, p. 44) provided a description that clinched it with the theocracy ruled by leaders having a glowing aura:
"With Flaming Swords." Astounding Science-Fiction, September 1942,
109-130. A major war ends when a single L-ray shot from an electron
cannon causes terrible destruction over a large area. People nearby
have their germ plasm mutated and bear sons with a glowing aura. In
the anarchy following the war, the mutant children found the Religion
of the Saints, establish a theocracy, and rule benevolently for
centuries using advanced science to reinforce their divine authority
and enforce their will. One Saint scientist discovers the truth of
their origin and reveals it to the world.
Thanks to b_jonas for impelling me to take the effort to rediscover this obscure story, which stayed in my memory for over fifty years.