How does L deduce that Kira needs a name and not just a face?
Just reread the manga and it turn out it was the police, probably with L help, that concluded the fact that Kira need both name and face
from Death Note Chapter 11 One, page 5

Matsuda: He needs to know names? Is that true? I've heard he only needs to know the face
Chief Yagami: Criminals whose names were either wrong or kept hidden survived some mass killings, so the headquarters came to that conclusion


L didn't do the stunt on tv with Lind L. Tailor to prove that Kira needs both name and face. But rather to prove that Kira exist, can kill people without being nearby, and live in Kanto area in Japan.