Nope, I'm pretty sure there's no such reference in "Bread And Circuses."
Correct. "Bread and Circuses" confirms the three world wars mentioned in "Space Seed" which is no surprise considering Coon wrote/co-wrote both. Kirk's mention of Earth avoiding a nuclear holocaust is supported by Spock's statement in "Omega Glory" that the Yangs and Kohms, in their post-apocalyptic world, "fought the war your Earth avoided". Populations can be bombed out of existence ("Space Seed" with conventional weapons and Col. Green's series of genocidal wars need not have been global or nuclear. Most likely similar to, but larger than the ethnic cleansing seen in the Balkans during the 1990's.