Yes, murdering rhey was the key disagreement. The very first thing alicent says to aegon is don't listen to Otto about killing her. I'm not sure why you are thinking that was just a bump in the road, it's a fundamental difference in strategy.
Both parties wanted aegon because whoever had him could control the plot going forward. Beyond that, it's tough to do a coup without the person you're claiming is the new king.
No one was looking to murder him.
He only told rhey about it and literally said it was only passed down from king to heir. No way alicent knew. She thought he was talking about his own dream where his son was born with a crown.
There are white haired white kids who are alicent's, brown haired white kids who are rhey's, and white haired mixed race kids who are daemon's. I have to be honest, if that's confusing for someone then maybe they should go back to watching network television.