the warp core requires a matter/antimatter reaction.
I can't remember, did FC mention a warp core for the
Phoenix? That would be a bit of a continuity rift there. That was supposed to be new tech for the
Galaxy and onwards - it just keeps getting retconned back to TOS, though it shouldn't be.
Anyway, on the similar tangent, M/AM was also supposed to be relatively new tech for the TOS era, with most ships relying on fusion (technically, 'hard fusion') engines. This is why anything NOT a starship is damn slow, or has fuel issues, etc. The
Enterprise was supposed to be the tech-equivalent of the then-new nuclear carriers in the US Navy.
Either way, it seems more likely that Cochrane was salvaging a fusion-system of some sort for the power core. I have no idea how he could manage to have a high-energy ECM container for AM planet-side.
Keep in mind, this is another issue where TOS explicitly said Earth
did not have the nuclear holocaust. WWIII happened, but not with nukes. Makes it even more of a mess, doesn't it?
As for the presence of the bussards? No idea there. Cochrane
may have thought to capture deuterium once the launch was underway, I suppose. The bussards don't
have to get anti-deuterium. The plan old normal isotope will do quite fine for a fusion furnace.