Thank you for the answers so far. Let me try restating my problem. (Btw, I'm used to seeing the invariance of the interval proved using the Lorentz transform. Landau is taking a different approach, going straight for interval invariance armed only with the relativity principle and a maximum...
I would agree if we already knew that the same a applied for all infinitesimal intervals. However, at this point in the argument, the coefficient a applies only to one infinitesimal interval (dt, dx, dy, dz). This infinitesimal interval has a direction.
I just decided to look at Landau & Lifshitz' Classical Theory of Fields (English version, 4th ed), and I am a bit embarrassed to be confused already on page 4&5 of this book. The book can be viewed on archive.org.
The goal of this section of the book is to show ##s = s'## starting from only the...