True on all counts, but, was that engineering section in the primary or secondary hull? The location of "Engineering", AFAIK, is still open to debate.
As far as I can tell in TOS, there were at least two engine rooms.
In "Naked Time", Scott calls to "engine rooms" and in "Omega Glory" we see two engine rooms (one occupied and one empty) on the sister ship, Exeter.
In "The Ultimate Computer", we see the crew walk down the length of a circular corridor to an engine room (the only time we see this combination) which can only fit in the primary hull.
In "Day of the Dove", the entity leaves an engine room and departs from the engineering hull. Other instances of the engine room we only see a short corridor that can fit in the engineering hull.
So, at least two engine rooms, one in the primary hull and one in the engineering hull
AARRGH! The small details escape me quite often.
You're right, two distinct engine rooms. My question would be then; is there any point in having an independently controllable secondary hull in TOS? The weapons are all on the primary hull, so it would offer no tactical advantage and the primary hull has more than enough room for the entire crew to serve as a lifeboat. So would there be a reason to have a crew there to pilot it after a saucer separation?