People regularly accuse the police of being corrupt and stupid when the truth of the matter is the issue is simple indifference. The actual problem is that they are tasked with an unreasonable job with a contemptible performance metric.
The job of the police is simply to arrest someone. That is how they close a case. In the instance where they have someone in an interview room or stop someone on the street or a traffic stop they are looking for anything that they might be able to prove to a degree that a prosecutor could convict you. It doesn't have to be true. They honestly don't care. The majority of crimes are committed by repeat offenders. If they happen to arrest someone innocent thats ok because the guilty party will eventually be arrested for something else.
The defect in the OP argument is that he fails to realize the performance metric that police are measured by. And he clearly fails to reach the conclusion that sending the innocent to jail or prison is bad for society.
The 5th amendment is one of the sole protections that allows the innocent to protect themselves in a situation when the actual guilty party is not known.
The police and prosecutors lose no sleep regarding getting someone innocent to confess to a crime. Even if there is actual evidence that conflict with statements made in a confession. People like the OP watch too many technical procedural crime dramas on TV. Crime dramas that lead people to assume that the police and prosecutors will only arrest and charge someone if all the evidence in a crime fits the person they decide to accuse.