To take it a step further, if that was the forecast 4 days in a row, you would expect it to rain one of those days.

Not necessarily.

There is a 0.750.750.75*0.75 or a 31% chance of no rain at all

I'm not saying it's certain, but it's what I'd expect. It raining on one day out of 4 is the most common case.

No, but it means that the weatherman isn't particularly effective at forecasting the weather.

No it doesn’t, and this is a fundamental misunderstanding of how probabilistic forecasting works. If it rains 9 out of 10 times a weatherman says there is a 30% chance of rain, they are a bad weatherman, but they aren’t much worse than if it rained 0 out of 10 times they predicted a 30% chance of rain. A weatherman accurately assessing the probability of the weather forecast would see it rain around 3 out of 10 days they say there is a 30% chance of rain.

Only if they say that every day, and it rains every day. If the most likely outcome happened every time, then the model is likely wrong/under-confident. The prediction is never going to be 100% accurate until the event is happening/has happened. Up until then, there's always a chance you're wrong or something can change. Being wrong once isn't necessarily a sign that the whole system is messed up.