to be more precise, OP says L should use the fact that Light selects criminals only when they appear on Japanese TV.
In the manga, L did track down earliest of murders to Japan and therefore started his "world" broadcast from one region in Japan.
Light got angry at broadcast's message.and lost his cool, and fell into the trap of "world broadcast"
> Selecting criminals could be based on internationally accessible periodicals that plausibly every human has access to, such as the New York Times, and deaths could be delayed by months or years to broaden the possibilities as to where the Kira learned of the victim (TV? books? the Internet?) and avoiding issues like killing a criminal only publicized on one obscure Japanese public television channel. And so on.
> Worse, the deaths are non-random in other ways—they tend to occur at particular times! Just the scheduling of deaths cost Light 6 bits of anonymity