Giving a probability of one event only implies we think that particular event is possible. It doesn't say anything about what other events we are considering, so there is no necessity to describe the entire space of possibilities.

Just because you don't care about measuring other probabilities in the space doesn't mean that you can ignore it. If you don't know what the space is, it's like taking a blank piece of paper, putting an "x" on it, and saying that's where the treasure is buried: not only do you not know the territory, but you don't even know enough about the map for that "x" to have any value.

I think most people naturally adopt a Bayesian perspective, so I'm not sure what the problem is.

I think you're giving too much credit here. Go out and slip into casual conversation a remark about the probability of something and see how people treat it. You could be right about the human brain, though, and maybe it's really a First World problem created by "numerical literacy" education in schools to try to help people read the news. Every time they hear a percentage they think of the frequentist interpretation they learned in school.