It was a great way to handle issues while they were still small and manageable.
Someone unhappy on a project? I'd do my best to talk to the project mangers and voice the criticism in an impactful way, or possibly reassign the person.
Problems at home? I would have never picked that up during our normal interactions.
Those meetings gave me the opportunity to help my coworkers with lots of stuff they would otherwise not have mentioned.
It was a true pleasure when I was able to help them out, and even when I wasn't, I think it helped a lot that they just had that chance to vent and then get on with stuff.
While some one-on-ones might have been a waste of time, I believe that overall they had a very positive impact.
Reading the rest of the comments here, I think I might have been in a very favorable position. I started as a developer, and then got promoted at a time where I was the obvious choice. So I always felt like "one of us" during those conversations, and I could very much relate to every work-related challenge.