I'm on board with most of this... although (speaking as a parent) I don't agree that Lyanna would either want to, or expect Rhaegar to be okay with the idea of naming his newborn son after his (now dead) son.
A few of the YouTube comments also make a good point that it's possible Lyanna didn't know that Rhaegar's son was dead at the time, so it would be strange to then have the second son also be named Aegon.