From the federal rules of evidence:
> (a) In General. To satisfy the requirement of authenticating or identifying an item of evidence, the proponent must produce evidence sufficient to support a finding that the item is what the proponent claims it is.
https://www.rulesofevidence.org/article-ix/rule-901/
Specific to recordings
> An original writing, recording, or photograph is required in order to prove its content unless these rules or a federal statute provides otherwise.
https://www.rulesofevidence.org/article-x/rule-1002/
Unfortunately, the rules of evidence are pretty vague about what is actually required to prove that a video is admitable as evidence; but the burden to do so is absolutely on the party introducing it.
If you have that (absent other possible evidentuary concerns), it is up to the trier of fact (e.g. jury) to determine if how much they believe the evidence.
The problem here is the prosecution cannot meet even that low bar, and instead is relying on the various exceptions to the requirement.
Rule 902 lists evidence that is "self authenticating", and videos are not on it.
At that point the reliability of the evidence may of course be called into question, as with any evidence or witness. The question mark would be whether it was convincing.
As such, the question of deep-fakes should be relatively irrelevant to (criminal) courts. If a witness claims that a particular video was filmed on a particular day in a particular way, that is as much evidence as you need. Whether that witness is lying remains to be proven during the course of the trial.
Things may be different in civil court, where the standards for evidence are more lax I believe.
Prosecution has a freer reign during deposition and can play a video from Youtube[1] and ask the subject of the video if they were at the location at a specific date and time, and if the words are their own.
1. With the courts permission.