What the IAF did in Kargil is what the CDS is referring to.
And what the IAF did in Balakot is what the army wants the IAF to do.
What the IAF wants to do is something else entirely, one of which is sending the enemy back to the stone age by destroying its infrastructure. But this would mean instead of killing enemy soldiers and making things easier for the IA, the IAF will be using their resources for something else. So the IAF needs to find the balance between the two.
While the army is a tactical asset, the air force is a strategic asset. The army's thinking is based on the present whereas the air force's thinking is based on the future. For example, the army wants to capture a hill or take a bridge for immediate gains, whereas the air force wants to make sure you do not have power for the next 10 years or they want to destroy an R&D facility so you don't churn out more IP etc.
While the IAF's mission is important, they also need to make sure that the army is sufficiently supported during war. Otoh, through theaterisation, the army is making sure that some assets are guaranteed for their use because the IAF is much more interested in doing everything else but supporting the army.
The IAF has a history of denying support to the army. For example, throughout our history, they did not make any technological investment into providing CAS in mountains. When support was asked for during Kargil War, only the Mirages were up for the task but were used for much more important missions and the attack helicopters were insufficient. Hence the need for the army to push for their own air assets that would suit their needs. This problem is completely of IAF's own making.
Similarly the IA needs guaranteed CAS. They have 4 SCs now, which means they need 4 fighter squadrons dedicated to CAS. Within a decade, we may get 5 SCs, hence 5 squadrons. Giving the army control over some air assets assures them of air support and not leave them to the whims and fancies of an air force commander who may not necessarily understand the ground situation. This will leave the air force to achieve its core mission.