You mean, if they had documented the marks on Gabby's face and taken her statement that Brian had "shmooshed" her face the way LE in other places do? Because, where I live, I do believe both of this couple would have been written up (at the very least). Gabby ought to have been offered the resources that Moab has for women who are victims of DV. This might have involved taking both of them down to the station, which of course, the police wanted to avoid (and did).
Do people change behaviors after they are taken to police stations and a report made on DV? IME, yes. Yes, they do.
Would Brian had still gotten enraged and strangled Gabby? We don't know.
But she should have been made to put the DV hotline number in her phone, in any case. It's sobering when the police ask a person to do that - they both should have had to do it (I assume Brian had a phone).
We will never know, but that's what the policies are locally, where I live. DV hotlines (usually staffed by volunteers - many of us on WS have done this work) really do help.
IMO, and IME.