I think your statement might only apply to 32 bit (one of the constraints mentioned early in the blog post was 64 bit).

But even if it was 32 bit, then we would't have to copy a 1, since the syscall number for sys_write would be 4 instead of 1.

I get the same total size with both variants in 64 bit mode.

  push 1
  pop rax
  mov rdi, rax

Assembling to 48 89 C7 (3 bytes)

seems to be same in size as

  push 1
  pop rax
  push 1
  pop rdi

Assembling to 6A 01 5F (3 bytes)