If you have major concerns, why not just schedule a meeting? Why are you waiting for a monthly catch-up to air them? I think this whole one to one thing is pointless - if there's a problem, organise a meeting to discuss it (or talk about it during one of the regular team meetings if it's not a private matter); if not, let's just get on with doing some actual work.
I think the same way you do but I’ve noticed that many people won’t bring up issues unless you take them aside and ask the right questions. No amount of openness and trust seems to work as well as gentle prompting.
Of course, you need to trust your manager. I’ve made the mistake of doing that more than once.
In a department I worked in, there were no scheduled 1-to-1, just a weekly slot that was guaranteed to be available if requested. Nobody ever took it. The manager thought it meant that everything was fine, while in reality he had no idea about how people were doing or how they were feeling.
Nobody wants to come to the manager with problems when he's the one with the power and they don't have a trusty relationship with him. And you don't get trust if you don't communicate regularly.
I see what you're saying but it doesn't sound to me like the scenario you describe would be fixed by scheduling 1-to-1s. Sitting down to talk with someone you don't trust doesn't make you trust them - trust is earned through behaviour over time. If I don't trust my manager then our 1-to-1 is going to be pleasantly bland and provide no value for anyone.
Once the trust is there then the 1-to-1 is proven to be unnecessary because the employee will feel comfortable raising issues as needed.