What they really need is a customer service team that finds these people that are having genuine issues that can be resolved. They'd need the means to reach out and communicate without that person being able to subsequently share contact information with the world.
That said, I don't know what I'd do as a Google employee looking at this guy's situation. If david.smith@gmail.com is dormant and my name and ID say David Smith why should that give me access to that mailbox? How do you verify the contents without violating someone's privacy? We're not asking some tiny ISP to take a quick peek, this is Google and they have a responsibility to all of their users to practice great care. So we can't assume they're telling the truth and neither can Google. This guy's situation sucks but the mistake of turning over the account to the wrong person is catastrophically worse than the bad experience of denying the rightful owner access. This exact situation is why they have recovery options.
I think it's likely they're telling the truth and I do feel bad for them and their situation. I've been there myself with a yahoo account years ago but at the end of the day the only way I could recover it was using something I no longer had. The things in that mailbox that I wanted access to were also things I would be furious at Yahoo for sharing with someone else no matter what their identification or birth certificate said.