An interpretation of canonical facts, with the ideas and suggestions of others added in.

I.Preamble
Between 2063 and 2240, starships were equiped with pulse fission engines to power the warp drive engines. The nuclear reactors were equiped with energy shields that contained the positive and negative elements, a baffle plate or plates as a partition to separate these two kinds of elements from colliding, and circuits which were used to control the process of the uniting of the two elements. Unlike starships built after 2240, fission reactors were not used to power the impulse engines, Rather, the impulse engines were fueled by chemical processes.

Several issues arose from this type of warp drive.
1.) First, there is the engineering. Baffle plates, which were intergral to the drive engines of a starship, were prone to fractures suffered by the stress of warp jumps. If the plate or plates were not maintained, a plate could be destroyed by a fracture point. Such an event could lead to an uncontrolled mixing of two dissimiliar elements. This could result in loss of ship and crew.
2.) Second, starships were not fast. Faced with the limitations of pulse fission reactors and the fear of being obliterated by a ruptured baffle plate, starships crews were forced to travel at lower warp speeds.
3.) Third, as a result of lower warp speeds, starships would travel months or years to the nearest star system. This situation created what became known as the time barrier. The time barrier referred to the amount of time that a crew could be on a mission before supplies ran out. This time barrier was finally broken in the 2240's by starships using matter/antimatter reactors. (Intrepatation based on canonical facts from: "Charlie X", "Court Martial", "The Menagerie", "Up the Long Ladder", and First Contact.)

II.
Canonical facts about the USS Republic
"Court Martial"
Commodore Stone: It's common knowledge that something happened to your frienship.
Captain KirK: It's no secret. We were assigned to the same ship some years later. I relieved him on watch once and found a circuit open to the atomic matter piles that should have been closed. Another five minutes, it could have blown up the ship.
Computer: Ship numenclature-specify.
Captain Kirk: United Starship Republic, number 1371.
Commodore Stone: Continue.
Captain Kirk: I closed the switch and logged the incident. He drew a reprimand and was sent to the bottom of the promotion list.

The Undiscovered Country
A listing in an okudagram

An interpretation of canonical facts
The USS Republic predated the Constitution Class which came into operation in the 2240's. The class to which the Republic was a member had to be commissioned at about or after the Class J starship. (A historical note in our world: US navy submarines were identified by alpha designations from 1903 to 1933, from A to S, with no P. This is to show that I am not on the bend when I say that the original classes of starships were identified by a alpha designation. ) Her class carried a alpha designation. This class employed pulse fission reactors for the warp drive engines.

In 2255, Ensign Kirk and Lieutenant Benjamin Finney were assigned to this starship.
Digression
How did I get 2255 for this assignment, when the Chronology assigns 2250 for Kirk's service aboard the Republic?
My answer-
2252 Kirk a midshipman at academy ("Shore Leave"-Kirk mentioned as a midshipman who was tormented 15 years earlier by Finnegan. Episode set in 2267. 2267-15=2252.) This year, he met Benjamin Finney.
2255 Kirk was promoted to lieutenant. "The Corbomite Manuever". McCoy says that Kirk was promoted about 11 years before 2266 to lieutenant. First assignment as a lieutenant-field study on Tyree's planet. ("A Private Little War", 13 years before 2268.)
2256 Kirk graduated from academy. Assigned to the USS Farrugut as first deep space assignment.
2257 Kirk survived attack by "vampire cloud" ("Obsession", 11 years before)
In the above conversation, Kirk said the incident occured "...some years later...". If Kirk had said a year later, this would be 2253. If Kirk had said a couple of years or two years later, this would be 2254. He didn't say either one. This leaves 2255 as the year of the assignment. 2256 is not a possibility for Kirk was a lieutenant at that point in his career. (And I haven't forgetton that Kirk mentioned in "Where No Man Has Gone Before" that he knew Mitchell fifteen years before, 2250. However, a point needs to be made. In that episode, Dehner says that Mitchell and Kirk didn't become friends until the former joined the academy. At the time of Mitchell's joining Starfleet, Kirk was a lieutenant and instructor at the academy. This places Mitchell's joining the academy after 2257.)
Returning to the topic, the USS Republic was assigned to Federation space and not on deep space exploration.

In 2293, the USS Republic was on active duty.

III.
Is the USS Republic ("Valiant") the same as the USS Republic ("Court Martial")?
I believe so.
Reason-
USS Republic of DS9 hadn't left the Sol System in 50 years, or since 2324. If the USS Republic of TOS was commissioned before 2240, this would make the ship well over a hundred years old by 2375. A ship this old couldn't be modified to serve in the modern Star Fleet. However, with her very basic technology as compared to contemporary starships of the 2370's, she could serve ideally as a trainer, just like the US Navy used ships from the 1800's to train sailors. Furthermore, there was very little time for a second USS Republic to become obsolete by 2324. Most ship classes-Constellation, Excelsior, Miranda, Oberth, and Sydney-that have come from the 2280's and 2290's are used by Starfleet for defensive and offensive missions, cargo missions, and survey missions.

And, this supports Picard's statement in "Relics" that there is only one Constitution Class starship in existence in 2369. The only other known Constitution Class starship in existence in the 2360's was lost two years earlier, in 2367, at Wolf 359.

IV
Summary of topic

Based on the description of the technology in "Court Martial", I reasoned that the USS Republic was a starship commissioned before 2240 of unknown class and served Starfleet on active duty to 2324. In 2324, she was refitted to be a cadet ship for training. Fifty years later, she continues to operate in that capacity.

However, I'm not quite convinced a modern starship couldn't have "atomic matter piles" in addition to the main antimatter powerplant. So the Republic need not necessarily be solely powered by antiquated systems, even if she predates the Constitution class.

Also, I'm not really fond of the idea that early impulse engines were chemical rockets. LaForge does say in "Relics" that these engines have not changed much in 200 years, or after the 2160s. Pre-Romulan-War reaction engines may have been chemical, but the Daedalus class should have had the current fusion/subspace system already (else where are the gigantic propellant tanks?).

There is another possibility. Could the pulse fusion reactors power both the impulse and warp drive engines?
The only designs seen from the period 2063 to 2240 are the Phoenix (2063), SS Mariposa NAR-7678 (2123), and USS Horizon NCC-176 (2168). These ships have no discernible impulse housing as can be seen in the Constitution Class and later starships. Nacelles of the USS Horizon, as well as the Bird-of-Prey of 2266, are very long in proportion to the hull and thick with vents located at the rear. Could the nacelles housed both impulse/warp drive engines of the USS Horizon and the impulse engines of the Bird-of-Prey? I believe so, and I believe that LaForge was referring to the design and placement of pulse fusion reactors in starships.

The Constition Class starships USS Constellation and USS Enterprise give evidence that impulse engines were undergoing development in the 2250's and the 2260's. Oldest known of the Constition Class starships, the USS Constellation NCC-1017, had the impulse housing. The surface of the housing was smooth with no impulse vents. She may have contained her impulse engines as the later Defiant Class did. By the 2250's, the USS Enterprise had the first impulse housing design on record. This housing seen in "The Cage" and "Where No Man Has Gone Before" had nine holes going from port to starboard. By "The Enemy Within", the USS Enterprise impulse housing had been replaced with a newer model. This housing had two trapezoidals on either side of the housing with a chevron in the middle. Further, at this point, the vents at the rear of the nacelles were capped with white domes. Some kind of design improvement had to occur for these changes to occur. One suggestion that comes to thought is the separation of impulse engines from warp engines.

If we accept that there are older classes of starships based on technological grounds, then there are at least five classes of starships operational in the 2260's: Enterprise-type, Antares-type, Republic-type, Class J type, and Columbia-type. Of these, the Enterprise-type is the most recent class and the Class J type is the oldest class.

My understanding of the dialogue comes from reading the subtitles, which are very accurate.

Furthermore, few of these older ships were called "starships" at all. But that's just nitpicking.

As is the mention that there is no proof that the Enterprise herself did not have baffle plates and atomic matter piles.... And I'm sure it did have "circuits".

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if you here me talking on the wind...