It's kinda basic, really...in order to prove your case, you've got to prove that item A, be it weapon or contraband, belongs to the defendant. You'd think somebody'd take the time to dust for prints, rather than just expect the jury to take the prosecutor's say-so.
It's not enough to have cops testify that they personally saw him throw it out the window?
Exactly.
It should be obvious that you can't get good DNA evidence or CSI-quality stuff on most, or even many crimes.
But you should expect at least some level of basic competency on the part of the investigators to link the murder weapon etc. to the defendant.
Assuming a hamburger belongs to the defendant (it sounds like this was the only piece of evidence to tie him to the scene, so they just assumed it was his) does not cut it.