The other interpretation is that Kirk and everybody else knows the problem with antimatter is always with the warp core, and therefore the core is not mentioned by name. Kirk just wants Scotty to disengage the nacelles from "it", then jettison "it" if he possibly can. Nacelles being the part of the power loop that consumes what the core puts out, meaning the loop will remain "live" unless the nacelles are disengaged from it first and you can't jettison a core from a live loop lest the ship blow up.
Or, replace the core in above with the antimatter tanks, because thar's yer problem in the most literal sense. Jettisoning the nacelles ought to solve nothing much, and indeed if that's what Kirk tries to achieve, then why's he micromanaging this "disengaging" bit first when it necessarily must precede jettison? It seems as if disengaging will already help some, and the jettisoning part is a nice extra they can live without.
"That Which Survives" has other relevant stuff, with Scotty struggling to jettison a single key piece of machinery from within the ship proper lest the antimatter blow. The struggle need not indicate lack of ejection systems, as we're observing a skilled saboteur at work, meaning all sorts of helpful stuff would be jammed and broken.
"One of Our Planets is Missing" has our heroes kickstart the warp engines by feeding antimatter (of a new, "regenerating" sort they just discovered) into a doodad in one of the two warp nacelles. Kickstarting systems rarely correlate with regular ones much ITRW, though.
Core ejection is hardly a mandatory system for TOS in any case - none was ever mentioned or shown in action, and the preceding ENT shows a ship that cannot eject anything without people first spacewalking to unscrew a dozen bolts. But ENT also firmly estalishes that "warp core" is a vital piece of starship machinery, and ST:Beyond supports the notion that those things were fashionable in the 2140s already. Having a core sounds like a must for Kirk and Scotty; having it eject at the push of a colorful button, more like an optional extra.
Timo Saloniemi