In short the Kiev class was build to counter NATO ASW forces.
As such the fighters are there to provide aircover for the fleet, but not strike capability like on western carriers. That role is taken up by the missiles.
Exactly. The 1143s were designed (like the 1123s) to defend the boomer bastions against NATO ASW groups and subs hunting soviet SSBNs. STOVL fighters on ships don't start to become major air threats until the deployment of active radar AAMs like AMRAAM, R-77, and R-27AE.