I would agree that in a domestic violence situation in the US, the police are more likely to believe the woman than the man. That said, the second sentence of your PSA is just a specification of the general rule that criminal courts tend to defer to police judgment in practice. Your last sentence is useful practical advise, at least right now (unfortunately).
In short, the second sentence is a specific application of a separate, gender-neutral problem with the way US society (particularly the criminal justice system) is organized. And the first sentence ("few legal protections against women") is false as written. If it is read as limited to DV situations, it adds no information above the latter two sentences. If it is not read in that limited manner, it is easily falsified, even in the criminal context.
Don't assume because I didn't provide more examples of how this information applies doesn't mean there aren't any. That was deliberate. If you want a hint, you can look into gender sentencing bias, plaintiff gender bias, and maybe, to see the far end of it, how courts have ruled on child support payments from statutory rape victims when the victim is male - those aren't the only cases when rape victims have paid child support to their rapists, but they're the easiest to find and the least horrifying to consider. You probably don't want the full picture....
P/S/A: There are single sentences which can create life-changing amounts of difference.