From a research-based perspective, there is evidence that proves the interdependence of all three memory systems: the sensory, the short-term and the long term memories.
This entails that all three systems of memory are equally important because each of them fulfills a specific role on their own, and in combination with other systems. Moreover, the neural structures that take part of the memory storage process are different from each other, but equally significant when it comes to memory and information retrieval.
Sensory memory is the sum of all input, as we hear or see it. This is the primary starting point for information processing. It is obvious that no process can start without a starting point. This is what makes is so important.
Short term memory is also important as it contains immediate facts whose retrieval can indicate whether our brains are working correctly or not. If we are able to recall facts that have just been shortly learned, we are well on our way to long term memory storage.
Long term memory may not be "the" most important, but is certainly quite influential for the everyday lives of healthy individuals provided that they do not have any conditions that may hinder their cognitive processes.
To illustrate the influence of long term memory, think about this: procedural memory constitutes a series of repetitions and body movements that end up becoming an everyday habit. It is the same as "muscle memory", such as it happens when we ride a bike, or drive a car. The part of our brain that deals with procedural memory is the caudate nucleus, where Stimulus-Response connections are made consistently. This S-R dynamic is what motivates and demotivates our everyday choices and activities. Therefore, this system of memory is of extreme importance.
Then there is the declarative memory, which constitutes your main unit of storage in the brain. Having the hippocampus as the structure where it all happens, this part of the brain makes the Stimulus-Sensory connections (SS) that makes humans flexible enough to elicit memories at will, and use them for different behavioral situations.
Within the Long Term Memory, there are also connections between Stimulus and Affect which occur in the amydala. This monitors the way that we condition ourselves to behave and react as it creates the "sweet" and "sour" memories that often dictate our behavior.
Therefore, all three memory "storage units" depend on one another. Our human brain, however, has shown a capacity and resilience that makes it overcompensate for whatever input is not properly processed. It is all like a puzzle; if one thing is missing, the entire system becomes incomplete as a whole.