That flag with the arrow pennant - it does seem to be a recurring theme throughout the series, in the decals on the ships and the big reflective metal thing behind various Admirals at star bases. Is this design a big part if Starfleet heraldry? Has it appeared elsewhere? I'm genuinely curious about this.
Yes, this "boomerang" symbol appeared on the side of the ship hulls (both the Enterprise and the Galileo) and behind Starfleet admirals and the courtroom on "Court Martial". I've always loved the symbol and am interpreting it as an official symbol of Starfleet Command, before it adopted the Enterprise Delta as its official symbol, (sometime between TOS and TMP).
However, iconography in TOS was a mixed bag and there's no definitive answer as to what Starfleet's official symbol was. It took them half of season one to even nail down the name "Starfleet".
Starfleet Commodores and Admirals in TOS uniforms all had a "flower-burst" patch sewn into their uniforms, which could also be interpreted as Starfleeet's official symbol.
We, of course, see different symbols on the uniforms of other Starfleet ships in the series (The Constelation, The Exeter, The Antares all have their own symbols) and it is a common belief that Starfleet ships all have their own symbols prior to TMP.
However, it should be noted that the more familiar Enterprise Delta also appeared on non-Enterprise crew members uniforms throughout the series (see the bar scene in "Court Martial", the scenes onboard the Defiant in "The Tholian Web", the officer oogling an Orion Vina in "The Cage/The Menagerie").
Bob Justman even wrote a memo to the production staff after "The Omega Glory" demanding that the crew stop using diffeeent symbols for other Starfleet ships, as it was his belief all Starfleet personnel should wear the Enterprise delta. Looks like they paid attention for "The Tholian Web" as the crew members of The Defiant all have Enterprise deltas on their uniforms, but then again, in another awkward move some 30 years later, a new unique patch was given to the crew members of the Defiant in "In A Mirror Darkly", which interestingly was a more angular version of the Starfleet boomerang.
So, it's up to interpretation in a few ways. I'm taking the following stance:
1) Each ship had its own unique symbol and crew members attached to these ships wore that unique symbol on their uniforms
2) This pre-TMP boomerang shape, since it appears on the side of ship hulls and behind admiralty, is the official symbol of Starfleet to be used on wall decorations, seals, flags, etc.
3) The uniforms of Starfleet Command personnel, and not attached to the operation of any one specific ship, will feature the flower-burst patch.
To me, this is the most logical way to sort out some of the inconsistencies.
