Purpose of repeating a 16-byte key for Salsa20
If we fix the second half of the key to all zeros we have at least 2 potential issues:


The zero-bits offer no additional diffusion effect, whereas repeating the key increases the diffusion effect.
Normally the adversary would know $1/2$ of the input and control $1/4$, but with $16$ more static known bytes, the adversary now knows $3/4$ of the input. This might give the adversary enough control to mount more attacks that may have not yet been analyzed; though likely only with a reduced round chacha.


Without doing the expensive analysis, I can only conjecture that the static zero bytes would only weaken chacha. I do not know by how much, but I do doubt it'll be enough to break the full 20-round chacha.