I think this is essentially the divide that separates frequentists from bayesians. I'm sure someone can chime in and correct my summary if it's wrong, but broadly speaking, my understanding is that frequentists consider probabilities to only make sense in the context of a repeatable event where you can measure how often something occurred, whereas bayesians consider probabilities to represent the "likelihood" of something happening even in a one-off event, which often is conveyed in terms of whether you'd be willing to take a given bet with certain odds; if a bayesian thinks something has an 80% chance of happening, they'd consider it a fair bet to offer someone $4 if it does happen but get $1 from them if it doesn't, since 80% is four times more likely than 20%.

The downside of bayesian probability that you point out is that isn't not straightforward to evaluate whether odds were correct or not. An example of this would be the 2016 US presidential elections; even the highest estimations of the odds of Trump beating Clinton were 30%, which after the fact was often cited as the predictions being "wrong". It's not easy to falsify this though, because a 30% chance doesn't mean something is _guaranteed_ not to happen; maybe it really was 70% likely not to happen, but we happened to end up in the 30%!

Frequentism doesn't suffer from this issue, but it also makes it impossible to talk about certain types of events (like presidential elections). In practice, bayesian probability gets used a lot when people want to talk about those sorts of events because there's not really any obvious alternative.