I don't think it's a "gotcha question" to ask how ideological claims are tied to reality in practice. But lets keep it high-level as you desire.
First, you describe a "key contradiction" that the capitalist economy as a whole would operate differently from a corporation. But it's unnecessarily reductive to demand such consistency. Ideologically a corporation is closer to an individual actor than to the overarching economic system imposed by government. In practice corporations grow from a single individual to huge conglomerates on the merits of their market performance, not from threat of violence, monopoly on employment, or other political power.
Second, you claim, "large corporations never make use of the mechanisms of capitalism, instead relying on central planning," which is a bold claim requiring evidence. There is tons of autonomy granted across the levels of a corporation, and a long history of conglomerates and franchises that cover the gamut of such regimes. If all of these are "central planning" then you need to explain what the alternative would be, otherwise its just a hand-wavy no-true-scotsman claim that can't be meaningfully examined or discussed.