Certainly those regions are doing
something in humans; and certainly those regions had those functions in species without a cortex. But the implication that these regions' presence in the human brain implies they're the ultimate arbiter of the same functionality there, isn't guaranteed to be true.
Brains really do use "subsumption" (in the https://en.wikipedia.org/wiki/Subsumption_architecture sense) to accomplish various features; there are motor signals that some earlier-evolved part of the brain would be emitting if they were the only thing "online" in the brain, that are actively suppressed or "overridden" by another, later-evolved part of the brain, often a part that only "comes online" later in brain development. (Thus "early instincts" that disappear during development, like the infant diving reflex.) Often, these subsumed neural processes reappear when the region suppressing them is damaged, as in the normally-suppressed human lordosis reflex, or the normally-suppressed-after-infancy suckling reflex in most mammals.
There's no evidence that I know of that the limbic system and the cortex are in this sort of relationship; I'm just saying that this kind of relationship isn't unprecedented as a thing human brains do.
If such a relationship were to exist between the cortex and the limbic system, then both regions could be said to be "in charge" of cognition; sort of like a television tuner and a VCR can both be in charge of the image being displayed. One state machine (the TV; the cortex), passing input through to another state machine (the VCR; the limbic system) only in some subset of its states (the right "channel"; the right arousal state.)