I don't doubt for a second that of all the forces in Afghanistan, perhaps only the Taliban are Pakistan's natural allies to some extent. And there's no doubt that they want good relations with us, and value our foresight in recent history and our tacit support.
However, one must weigh this issue from the IEA's perspective. The Taliban are not an entity like our army, uniform and disciplined, all answerable in totality to their chief and commanders. It's a much less cohesive entity made up of various tribes, groups, networks.
They have just taken Kabul and busying themselves with getting global recognition, keeping the state running, all while managing internal enemies. The US is no longer seen as their main enemy. The main enemy is IS-K. They are like the Taliban, but more radical. If the Taliban cannot keep internal unity, or if they disillusion groups from within their ranks, groups like the TTP, as well as other militants. They are understandably fearful that these groups will defect to join their no. 1 enemy. The Kabul attack is just a sign of the changing times, before Taliban were insurgents and the US + allies were governing, now the Taliban are governing and IS-K are the insurgents. Luckily for the Taliban, everyone hates IS-K, including US + allies, former Afghan regime, Iran, Pakistan, and other regional powers.
IMO there is a strong possibility that the IEA will not crack down on TTP out of fear of TTP types joining IS-K and causing the IEA more trouble. TTP elements have also signalled this to the IEA, some groups of theirs have joined IS-K, others have signalled that they may defect from IEA allegiance to IS-K if pushed. I'm not sure how worrisome or credible this is from the IEA perspective, but it's important to note. IEA will therefore will try to manage their internal affairs by balancing how they present themselves internally and externally. They have to convince their own men that they are still the old Taliban and therefore everyone should support them, while simultaneously telling the world that they're looking to co-operate.
Mullah Baradar met with the CIA director in Kabul a few days ago, what they discussed was the presence of, and threat posed by, a common enemy. The CIA shared intel of an imminent IS-K attack and offered assistance. The Taliban however cannot afford to be seen as selling out to the former occupiers for fear of dissention from their own ranks, and for fear of emboldening their new internal enemy.
This is also why they will treat TTP with kid gloves. It's why Khalil ur Rahman Haqqani when asked about the TTP basically said that Pakistan should negotiate with them, ie. don't expect us to fight them for you.
It's a tough situation, and we need not to delude ourselves with celebration about the IEA taking over. It seems everyone's safely assumed that the TTP, BLA/BLF, and all these other militants are done for. It's not as simple as this, and let me assure you, there are even more groups and individuals in Afghanistan besides the aforementioned who have been at war with the Pakistani state. Stay sharp and don't count your chickens before they hatch.