For Alicent it's three things.
First Rhaenyra didn't just lie to Alicent, she lied to Viserys. And the king after meeting with Daemon who confessed or rather didn't deny, decided to cover for Rhaenyra at the expense of Otto (her father).
Second Alicent found out that it wasn't Daemon who could've possibly tricked and or raped drunk Rhaenyra. It was Ser Christan, the two of them kept it secret and he even says that she started it, not him.
Based on what happened to Daemon she knew that telling the king wouldn't end the wedding plans. And she was right as Rhaenyra had already reached an understanding with Laenor.
(Alicent doesn't know if the king knew it was Christan and Daemon was also banished to protect Rhaenyra. After all she finds out there's plenty of gossip about Rhaenyra's late night adventure)
Third, Alicent would've probably been willing to concede the throne to a Velaryon / Targaryen heir as long as she and her children would keep their place in court.
However, she finds out that the heir to replace her son might turn out to be a Cole instead. All her efforts and sacrifices upholding the traditions and rules for both her sake and the realm would've been in vain.
Rhaenyra would sit on a throne of lies.
All she knows is that Rhaenyra is capable of lying to her best friend and possibly her father and is willing to throw "anyone" under the bus to save her own skin.
Her father told her that if she didn't challenge Rhaenyra's claim all she could do was plead her case with the soon to be queen and pray for mercy but she knows that Rhaenyra can't be trusted.
I didn't understand the Cole scene in terms of editing. Like it was very confusing when it cut from the two Targs close up and then a scream. Felt like it was missing a quick scene of what prompted that.
Cole has a conversation with Rhaenyra on the boat. His idea was that it was only a matter of time before she would marry him (as her intention was to decide for herself) and not some random lord.
Rhaenyra says that it doesn't matter because she has agreed with Laenor that their marriage is only political and they can each pursue their own lovers.
Cole however takes great pride in being a member of the King's Guard and Rhaenyra's protector. He literally says that's all he has to his name. He's willing to give that up but only if it means building a life with Rhaenyra elsewhere. The prospect of being her toy boy doesn't sit well with him. He's quite torn about having broken his oath and then he finds out that others know and they can use it to have control over him and Rhaenyra.
So he decides the best solution is that he and Laenor's lover are both removed from the table. So Rhaenyra is protected from being blackmailed (after all he is her sworn protector) and he won't tarnish his name further.
