From a Pakistani perspective: this is something long coming. We know India has been supporting TTP and the like for over a decade now, and has been responsible for dozens of terrorist attacks in Pakistan. The fact that the previous did not speak up about it is something that has irritated a lot of Pakistanis. We know the NS government had business dealings in India, hence he likely was putting his own personal interests first as opposed to Pakistani interests. We should have been shouting about Indian-sponsored terrorism (not just in Kashmir, but broadly the terrorism they support in Pakistan) for a long time, and finally we have a government that's doing that.
With respect to getting this done in UNSC, this can't really be compared to Hafiz Saeed or their ilk because they have been listed as terrorists as non-state actors. India has never been able to get Pakistan listed as a state sponsor of terrorism at the UN.
By listing these 4 as terrorists, you'd effectively be declaring India as state sponsor of terrorism. In this era, you basically have to have no friends (ala Iran, North Korea etc) to be listed as a state sponsor of terrorism by UN (that is different than US state sponsor terror list). Not gonna happen. India couldn't get Pakistan listed for decades. Evidence is mostly immaterial here. It's whose interests are yours aligned with and whom your enemies are.
But that's not really the point of doing this IMO. All Pakistan needs to do is beat the drum about Indian sponsored terrorism and repeat it where-ever possible. And do not mention Kashmir at the same time - just discuss the state sponsorship of terrorism India has perpetrated within the non-disputed territory of Pakistan. This is the same strategy India used (although now they over-do it, and their complaints are becoming less effective, so maybe there's a lesson to be learned there) and initially it worked well, so it's something Pakistan should take a lesson from.
When the investigative agencies do not have jurisdiction in the area where the proofs are, it becomes very difficult. In Hafiz Saeed case, it was not the open admission. Open admission was just the collaborative evidence.
David Headley testimony is part of the evidence. He was not in Indian custody.
Hafiz Saeed voice recording while monitoring the Mumbai terrorism attack is with us. We gave that recording to all countries and asked them to satisfy themselves about its authenticity. Also asked them to get recordings of his speeches for voice samples to compare.
There will be other evidence too. It is not possible to get someone to be named as an international terrorist without convincing evidence.
I think the main thing you guys are missing is that getting a non-state actor listed as a terrorist is different than getting a state listed as a terrorist sponsoring state. The former is what happened with Hafiz Saeed, and latter is what Pakistan appears to be trying to do here. In this day and age, a country has to be extremely isolated to be listed a state sponsor of terrorism. That's not the case with either India or Pakistan.
Basically, if you are a country that has at least a few influential friends, then you can perpetrate as much terrorism as you want and have nothing happen to you. Your friends will always back you at UNSC. I mean, Israel is a glaring example of this. They don't even have Europe as friend. And nothing happens to them at UNSC. Evidence is mostly immaterial here. It's whose interests are yours aligned with and whom your enemies are.