While we would hope that it settles the argument, it doesn't address the emotions. And therefore I suspect that it would just make people's arguments worse.
It's a Black Mirror episode, if you want a preview: https://en.wikipedia.org/wiki/The_Entire_History_of_You
The first Black Mirror episodes were really like looking at the future.
I really recommend to see this episode as it shows what consequences such devices can have on people's lives.
Portable microcassette records have been around for many decades (and hitting record on a phone to capture audio is something anyone can do without taking the phone out of the pocket).
There's a reason you're wondering about how well this idea works.
I try to get as much relationship decision making as possible done via SMS text or email.
And the moment there is a discussion/argument that turn to “what you said/what I said”, I say I’m only willing to talk about tangible facts, not conversation recollections.