As for core ejection, it's probably a technology that will take centuries to perfect yet. After all, we have no evidence Kirk's ship would have had anything like that available: if his power system started acting up, the best chance would have been to abandon the whole ship. Except he couldn't do even that, because he had no deep space -proof evacuation system in evidence.
The closest TOS ever comes to a "core ejection" is a cumbersome manual operation Scotty has to perform on the antimatter flow system in "That Which Survives": there appears to be an ejection system for the fuel stores, but it consists of Scotty himself installing makeshift explosive charges that will sever the antimatter (and the engineer!) from the ship, probably in a highly destructive manner but still one that will save some
It may well be impossible to eject all the components of an antimatter power system in a safe manner: there's always some leakage that, in the event of the ejection, results in local explosions and the total loss of the crew. Such a system is still worth having if it makes a difference between "starship becoming a gutted wreck" and "starship becoming a fireball that consumes nearby starships and wreaks havoc on the planet she's orbiting".
Regarding the original question, we unfortunately never got a detailed look at other TNG era starships, in or out of action. A few glimpses into old Constellations
suggested these ships were robust and easy to rig with all sorts of fancy 24th century gimmickry, such as a remote piloting system, a system for one man to run the ship for a short while, or a two-second warp drive. Picard didn't give praise to the design in "Relics", though.