There are alternatives to a human killer which would provide some opportunity to make headway, which do not have priors that are obviously lower than a human with a supernatural weapon, such as extraterrestrials or some sort of supernatural creature which is humanly beatable.

The first point where I got really pissed off though was when L jumps all the way to "the killer must know the victim's real name" based on the murder of Lind L. Tailor. Lind L. Tailor was a convicted criminal, and L. wasn't, and killing criminals was already Kira's suspected modus operandi. It was not just possible, but probable, that Kira wouldn't react to a non-criminal's threat to apprehend him (it could have been against protocol, against his/her/it's moral code, unnecessary because Kira is completely unassailable, rejected as unnecessary because Kira is confident enough to think he/she/it is unassailable, etc,) even if doing so was entirely within Kira's abilities. And even if we take for granted that Kira would want to kill L, and assume that Kira has magic action-at-a-distance murder powers, but not magic action-at-a-distance information gathering powers, then whether the victim's name is known or not is just one variable that's flipped between L and Lind L. Tailor. L could just as well have been impervious because he eats too many sweets.