Let's say that the software is published, and the code is audited and it looks ok - it seems to implement all of the intricacies of the transfer system and so on correctly. Then what? What if the operator forgets to use the latest version? or puts a different piece of software entirely on the counting system?
Having audited the source code really doesn't help; it won't remove the need to perform independent verification.
Similarly, merely publishing the input won't help much either; how do you verify that the published input corresponds to the actual votes? It won't remove the need for independent parties to observe the raw input (paper votes) and to make their own tallies; in which case those parties can publish their own copies.
Asking for the input to be published and the source to be published is just scraping the surface and won't add meaningful security. You need independent observation by mutually distrusting parties.