It would be extremely hard (probably impossible) to hide the fact that you need a criminals name and face to kill them. At most he could've gained a bit of time by being more careful, and at that point he didn't believe the death note would work anyway.
Maybe if you assume that psychic killings are possible from the outset. While people might be able to figure out that something is going on, proving that someone is doing the killings remotely via a notebook is quite a leap and only the crazy arrangement L tried made proving that possible.
