I think the police should have held out until they had more evidence -
Charge him with the other crimes he admitted to, but hold off the murder charge until he had some time to sweat in prison for the other convictions.
They may have had more information come through, or he may have cracked under the stress. Without a lawyer to guide his every move , I get the impression this is one guy that would have broken eventually.
I think he did it, but on the evidence provided the case was never going to get a guilty verdict. No concrete evidence, no proven murder weapon, no witnesses. The perfect crime perhaps.