Ok, these are my random thought about the whole thing.
I think House of the Dragon is like a chess game, two sides, two queens, various other pieces and a lot of strategy. I don't think either side is good or evil, they are just doing what they need to do to survive and win.
For many years Alicent was protected by the king but now he is gone she is stuck in the middle. She knows if Rhaenyra becomes Queen, then Alicent's children will be killed. I think she felt that she could somehow bring the whole situation to a peaceful resolution, so this week she finally made her power play.
Did she really believe that the king wanted to make Aegon his heir or did she just really need to believe that he said that? I think she heard what she wanted to hear because she was scared of what was going to happen and has now convinced herself more and more that it's the truth. I also think she thought that if she could capture Prince Aegon (another chess move) and she had Rhaenys' dragon then she could control the situation and bring it to a peaceful resolution, but she was wrong. Ultimately, she realised that Aegon can't be controled and that Rhaenys escape means that war is now inevitable.
So, I think, the search for Prince Aegon was fruitless and that it was meant to be fruitless. She ultimately did exactly what her father wanted her to do.
It's funny that no one really wanted Aegon to be king, but they needed him to be king to protect themselves. I loved watching Aemond and Helaena's faces during the coronation.
Why didn't they crown Helaena? I think being crowned was a huge moment for Aegon. Until now no one has respected him but suddenly he has the love of the people and you could see how it changed him and that change is scary. That moment was all about Aegon.
Why didn't Rhaenys roast the Greens? That would have been a huge mistake. Everyone knows she wanted the throne and by killing the new king she would be seen as taking over and taking power from Rhaenyra. I don't think that is what she wants. She will fly to Dragonstone and pledge her support Rhaenyra.
At least that is my interpretation of events.