^^^^ not quoting the entire thing
1) Dhritrashtra was never crowned because Pandu never retired as a king, he said he wants to repent for his crimes in the Van. In such situation how could there be another king crowned? Plus none of the children were born by then, there was no way Bheeshm could have anticipated how future dynamics would go. Dhritrashtra was therefore simply put on the throne as the caretaker because the king was alive and out only for penance, no one would have even accepted his decision to make someone else a king
2) Now this is very clear that Pandu wanted his son ro ascend onto the throne, that exactly is the reason why he asked for Kunti to go for Niyog (this is clearly mentioned that the intent was only to keep the stake on the throne) yet once Yudhishtir, Bheem and Duryodhan were born, Pandu didn't return, he didn't even send Yudhishtir back, what could be his reason? Anyone who is trying to ensure his child becomes a king will try to get him involved in politics and princely affairs since the beginning , but Pandu didn't do that, it clearly means that the Pandu his wives and children were not welcomed in the kingdom. Pandu in himself was a warrior its pretty sure that he had ensured the safety of his children in the forest where Bheeshm had limited influence owing to the area being under Brahmin control(such areas are not under the rights of the king, hence while staking Indraprasth, Yudhishtir clearly mentions Indraprasth excluding the areas of Brahmins). It was an unwritten agreement between the two, you stay away and your family is safe
3) Once Pandu was dead, Kunti realized that she needs to be in Hastinapur to ensure that her kids' rights are saved, she therefore gained sages support and came up with the Divine birth theory so make her children special and better suited than Dhratrashtras. Once she returns with such support and accepted story, Bheeshm has no option but to accept her. Not just this, by now Yadavs were also slowly gaining predominance and Bheeshm wouldn't have wanted to get a new enemy. So he let her in. Kunti had her goals clear, she took Vidur in her support. (If Yudhishtir was Vidurji's biological child then all the best, even if not, Vidur definitely wasn't happy there at being nothing more than a glorified servant in Hastinapur despite his talents and closest relation to the royal family). Bheeshm could not have done any direct attack of Pandavas that would have been suicidal to the image of the royal family (this is the reason I feel he had no idea about the Varnavrat conspiracy). Now they did plan internal attacks, but Vidurji anticipated and neutralised it
4) About the point why involve the kid, if he will blurt. We are talking of the princes who are taught secrecy, diplomacy etc. since childhood. There was no reason why Bheeshm would have feared this. He couldn't come to forefront. The major people designated to ensure the murders were Karna and Shakuni(,check KMG there is a full stop between Duryodhan n other two names), just that Duryodhan also gets involved since the matter affected him the most. Also he got exposed at the end. The narration was complied at the end of the war, by then Duryodhan was the official villian but Bheeshm was not exposed, he was still a respected figure, do you think Ved Vyas could have dared mentioned his participation without proof, forget civilians, even the other sages would have got against them. Yet his mentality and shrewdness is very much highlighted
5) The Pandavas had already lost their goodwill in the Rajsuya Yagya where a cousin, a friendly king who supported Pandavas campaign was killed by a non king. The Dharma wasn't followed in the Rajsuya by not making the eldest of the family the Pratham Poojya and instead giving the position to someone non deserving (at least public saw it that way) Pandavas didn't have that public and sages support post Rajsuya closing ceremony. So Sages didn't do much after what happened in Dice Hall, that is the reason Yudhishtir did his negotiations while at exile to ensure some support onto their side
Aside it was Yudhishtir who lost a fair game and whatever was done there was within law.
Kauravas did not lose their respect in front of sages. They freed Draupadi, gifted her more than she deserved or could have demanded for. For what exactly would the sages object? We find it wrong it is wrong but for people then, the act would have been taken as completely a benevolent one.
6)Your only assumption about Bheeshm not knowing about Karna is based on the fact that why didn't Dhritrashtra tell this to Duryodhan. Now I have two points regarding this
a) Dhritrashtra could never guess that it would help him, he would have rather taken it as a risk of Duryodhan losing trust on Karna and thereby probably his self confidence. I read your point about why Yudhishtir was not told since he might have gone into back foot, but that is how human psychology behaves they anticipate how would a thing affect him. While Krishnaji n Kunti would have thought that this would weaken Yudhishtir, Dhritrashtra would have taken it like it would shake Duryodhan. It's all preassumption they have to make, nothing could actually be a straight truth
b) If you think that Bheeshm Karna convo never happened because Dhritrashtra didn't say this to Duryodhan, then it is much more possible that Dhritrashtra got the news after the battle was over. Unlike what TV shows have, Sanjay wasn't having a live war Telecasted in front of him for which he was giving a running commentary. Instead he used to visit Dhritrashtra at the death of every CIC, tell him that Bheeshm, Drona, Karna, Shalya etc. are dead. Dhritrashtra used to cry at the state of affairs and then asked who all are dead/alive. Sanjay would give a brief of it, and then Dhritrashtra asks tell me in detail, and that's how he began. Dhritrashtra was definitely not at the war arena else he would have got the news of at least the death of the CIC. Now it's impossible without divinity for Sanjay to travel to Dhritrashtra after the death of every CIC and then return before the start of the next day war. It's pretty clear that this fragmented narration is a poetic liberty by Vyas, either Sanjay returned at the end of the war and gave all the details or Vyas revealed to us all the facts in form of Sanjay Dhritrashtra convo to make it interesting.
Dhritrashtra not informing Duryodhan (at a time when they were separated by miles) is no indication (forget proof) that the convo didn't happen at all.