I would think if Dalinar was lenient towards Adolin simply because he's his son, a lot of people would see that as corruption.
IRL, for instance, if a politician's son committed a murder and got away with a slap on the wrists due to the position of their parent, people would be outraged.
I actually think a lot of people would think Adolin's actions as outrageous, irrespective of Sadeas being a huge scum bag. Adolin basically decided to rage kill Sadeas, there were no special circumstances to argue such as self-defence. A civilised course of action would have been to gather evidence against Sadeas and then try him - then a death penalty can be administered.