I half-heartedly agree with the O.P. TOS had a nasty habit of sending the Enterprise's top officers into dangerous territory; but then, this was a built-in flaw with the show's format, since it was originally just focused on Kirk with the other characters being supporting roles. As the show progressed, it slowly (and probably accidentally) evolved into Kirk-Spock-McCoy and then a larger ensemble format.
"The Return of the Archons" did give us a tiny taste of seeing Sulu and O'Neil as junior officers sent to reconnoiter, and Kelowitz was seen as another junior officer doing reconnaissance work in "Arena", "The Galileo Seven" and "This Side of Paradise". But the "Bread and Circuses" story really demands Kirk take center-stage. So Roddenberry and Coon created a perceived problem right from the get-go.
"Bread and Circuses" is a tailored Kirk/Shat story, period. Unless you already had a character like Sulu or Kelowitz fully developed from prior stories, with plenty of lines / plots given to them, it would have been strange to just see Kirk sitting in his command chair telling Scotty to turn out Planet 892-IV / New Rome's lights, and Spock and McCoy having nothing to do for the entire story. So the show wasn't set-up for this during the actual TOS run.
Yes, it would be neat to see a character like Sulu or Kelowitz commanding the landing party, "The Return of the Archons"-style. I'm surprised none of the TOS-derived fan films have tried this to date.
As for the notion of Kirk taking redshirts down to New Rome with him, I disagree. No matter who would be sent down, the landing party in that situation should be small. No large, heavily armed contingent.
I'm not totally surprised by the criticisms raised here, but I really think there are more disturbing issues in "Bread and Circuses". "Patterns of Force" brought up the notion of McCoy injecting landing party members with subcutaneous transponders
to keep track of them in case they lost their communicators; this interesting and innovative technique was never used in similar situations in TOS again. That was a shame. "Bread and Circuses", "A Piece of the Action" and "Patterns of Force" failed to shed light on what happens with lost/stolen Starfleet-issue technology on a Prime-Directive-protected planet. What does a starship expedition do about those missing phasers, communicators, tricorders, or even medikits? This disturbing scenario occurred to me as far back as the 1970s. Thus far, no TREK series (or movie) ever explored this.