The Death Note characters being written as retards is besides my point
It's a public investigation everyone is aware of, it isn't a black-op. L announces he's working on finding Kira from the get go, which is a declaration he's working with the law in mind because of course he's being held to scrutiny by the entirety of Japan.
You do NOT let a Kira culprit go live his life unmonitored when you know you can actually catch him in the act since the act relies on him getting information on his victims
L was pressured by his entire team to scale back on supervision of them because of a lack of evidence, he knew better but obviously couldn't do much about it. Continuing to surveil them long-term was economically unsustainable, unmanagable and unethical and he was a private investigator so he was operating inside the boundaries of the law. The Kira investigation crew does not have unlimited resources, in fact they are a small team of about 10-12 people. L is a private investigator who funds the investigation himself and is therefore forced to consider the demands of his team unless they voluntarily leave, they're all there out of choice. L would absolutely agree with you, but he didn't exactly have many options at that point in time and this is made very blatant.
They have over 20 different pieces of evidence showing she sent the Kira tapes
All of them circumstantial. For instance, the stamps, the ink, and the notebook, all were simply similar but there was no way they could prove it was the same ones used to write to Sakura TV.