Yes of course, he would have punished himself mentally more than what the investigators might have dished out.
I know it's not your intention, but just the thought of a scenario where pilots are punished after the event as a lesson to others is scary.
If there is some actual incompetence - perhaps alcohol or drugs - then sure, some punishment is merited.
But far better to employ people with the mental ability and the experience to be an airline pilot, and to train them to do the job. Give them the resources they need. Test them regularly and pull their ticket if they aren't up to the task.
Depending on actual incidents and accidents as the driving force in pilot motivation is a terrible way to arrange things. The pilot shouldn't be thinking in a moment of crisis, Geez, I'd better do the right thing or they'll fine me and maybe throw me in prison. He should be thinking, here's the drill I've practiced, here's the procedures I've been taught, how can I make the best use of the limited resources and time I've got. Thoughts of punishment and future courtroom scenarios are the last distraction he needs at that time.
It's probably easy, once the forensics are in and all the facts available, to pick some optimum route through the crisis that might have led to a different outcome. But the pilot isn't sitting at a table with every fact laid out neatly, charts spread open, leisure to examine every alternative. He's got a few seconds of limited knowledge to make crucial decisions. Life and death decisions, in this case. And certainly there will be alarms and alerts and interruptions.
This is combat level, personal survival stress the pilot is going to be feeling. And pretty much coming into it cold.
From a Hollywood point of view, where the audience knows the ending already, it's a difficult tale to structure effectively. The climax is obviously Sully's crisis in the coughpit, but how do you fill the eighty minutes before that point without boring the audience rigid? Rearranging the story to give the investigation more prominence seems to be the solution chosen here, but I find that course a little shallow. Sullenberger did the best job he could, he relied upon his training, he achieved a good outcome. Nit-picking and criticising his performance afterwards for the sake of drama rings a little false to my ear.
But hey, a Tom Hanks movie is going to put bums on seats and eyeballs on screens, and that's what counts, right?