The only thing I’d quibble about is the ‘false’.
“It’s looking pretty bad for you at this point. I’m sure that you didn’t mean for it to turn out this way. Why don’t you tell us your side of the story. Maybe if you explain what really happened, we can figure something out.”
Then, when you type up the confession, put in a couple of mistakes. If the perp doesn’t pick up on them, point them out so the perp can make hand written corrections and initial them. That takes care of the ‘I didn’t read it. I just signed what they put in front of me and told me too.’
While ‘shut up and lawyer up’ is the smart thing to do, most criminals think that they can explain it away. And it will frequently get them a softer sentence for being cooperative.
There are definitely false confessions. They are not as common as the false “eyewitness identification.” (Most of the DNA exonerations have been on cases where there was a false identification.) But they do exist. The police tend to get a theory of “what happened” into their head fairly quickly on an investigation, and once they decide a crime was committed by a particular person, they zero in on them to the exclusion of all other possible alternatives. And the police are trained that they have to break down the psychological barriers of denial in a suspect. It works very well to get guilty people to confess, but when faced with the same techniques, sometimes they work on innocent people too.