Saturation or condensation is the first step in the formation of precipitation but not enough by itself to cause precipitation.
In general you have to have some type of lifting for precipitation and typically that is through air bodies being forced over mountains, warmer air being forced over colder air or through convection.
With low pressure like a hurricane, air rises into the atmosphere where it cools and condenses and results precipitation. The ground level wind speed is not required for heavy rain but a rapid vertical speed is. The extreme vertical rise in a thunderstorm is an example of a situation where you may have lower surface wind speeds but large amounts of rain.
In the case of a low pressure system like a hurricane ground level wind strengths would typically be associated with precipitation amounts.