If we could have civil partnerships for straight couples in the way that they do in many European countries, this wouldn't be a problem.
Doubtful.
Straight CP is rather a minority concern. Also in my experience, people who understand enough about marriage to object to it and also to want an alternative institution even if they do a bad job at understanding the dodgy history of CP tend to be more likely to understand what their position is as unmarried couples, and to have made at least some provisions. That's a generalisation, but it holds good a lot of the time.
The problem is people who don't know what the law is, or occasionally who do but CBA. People who think they'll be protected because they're common law, or marriage is just a piece of paper, or they don't need the state to recognise their love, or it's too much hassle. Straight CP would also be a piece of paper, would be from the same state that recognises marriage and would be the same procedure to book and attend. So they wouldn't do it.
Looking at the couple in this circumstances, she was very shocked to find she wouldn't be entitled to bereavement benefits. She'd evidently either assumed she would be, or not thought about it at all. Had a straight CP been available to her and her late partner, it wouldn't have solved either of those problems.
It's an interesting decision though. I wonder if the law will now change? The decision itself doesn't actually do anything for now. People were speculating about the benefit eventually changing to be attached to a particular child and going 'with' them, rather than being related to the dead person's NI contributions. But unless and until this happens, unmarried parents need to either marry if this benefit is important to them or make alternative provisions until they can guarantee it would be available.