It would help, but it’s not necessarily the prosecution’s burden.
If the defense asserts a Brady violation, it’s because the defense is asserting that there was some useful evidence that the prosecutor had but didn’t turn over.
To combat this, the prosecutor could either a) show that he did send it over (“Judge, my office emailed defense counsel on January 1st with a copy of the video”) or that the prosecutor never had it to disclose (“Judge, my office requested the surveillance videos, but the business never turned them over to us”). Note that the prosecutor is held to have access to whatever other law enforcement personnel have (so, the prosecutor can’t say that he never had it, although the police did).
But, while the prosecutor has to turn over anything they find that can help the defense, it doesn’t mean that they have to investigate on behalf of the defense. So, if there was some lead that the defendant should have followed up on, it’s on him to do so. Thus, if the defendant knew about some evidence but didn’t bring it up, it is likely to the defendant’s detriment, since the prosecutor is going to argue, “He knew about it, so it was up to him to investigate and introduce it in court.”
(Prime example: often, in the course of a case, the prosecutor will notify the defendant that they have received word that an officer involved in the case has a disciplinary history with the department. At that point, the prosecutor has done what it is required to do. It is now up to the defendant to subpoena the police officer’s records and/or request that the court provide information regarding private personnel information. If the defendant never does so, there’s no obligation for the prosecutor to introduce this information at trial, and if the defendant is later heard to complain that the officer could have been shown to be untrustworthy because of some disciplinary issue, then the response is that they should have been the one to raise it at trial. It would only actually be a Brady violation if a) the prosecutor never disclosed it, and b) the disciplinary issue was something that would have changed the case (usually, the fact that an officer had a PR issue with a fellow worker doesn’t impact their credibility as a witness)).