I think the Buddha was very wise in his statement that "Desire is the root of all sorrow." Sorrow, after all, is a feeling of sadness brought on most often by loss or disappointment.
I also think that human beings are motivated by desire, as you said. While this is initially counter-intuitive - in a tautological fashion, we might deduce the absurd conclusion that "human beings are motivated by sorrow" - it is actually a circumstance that can coexist with the Buddha's wisdom.
Why would someone argue that the Buddha's words make no sense? Well, although when we desire something and fail to receive it, we are unhappy; when we do receive something we desire, we are happy, as a simple function of evolution. Our brain must reward us for actions we take to secure what we want. So, desire is a mixed basket, a 50/50 that leads us to sorrow depending on the situation.
This isn't true, however, because even in the circumstance where a person receives what they want, they are only positioning themselves for more sorrow further down the line. I can see two possible scenarios after someone receives something they've been desiring: the first is disillusionment, and the second is anxiety.
In the first scenario of disillusionment, the person has become so fixated on the process of obtaining the thing or their own illusory perception of the thing that when they actually receive it, they are disappointed and disinterested. Think of Jay Gatsby in The Great Gatsby, who dedicated about 10 years of his life to forming a relationship with a girl he met just once before, Daisy. And when he finally managed to achieve this, after sacrificing his morality by getting involved with the mafia and seducing Daisy away from her (admittedly dishonest) marriage with Tom, he felt hollow inside. In a sense, he was experiencing sorrow, brought on by the loss of the desirous illusion he had fallen in love with.
In the second scenario of anxiety, the person actually does enjoy the item they've desired and obtained. However, they love this item so much that if they ever were to lose it, it would be a devastating blow to their psyche. In this case, think of a person who dreamed lifelong of being a parent and has finally had a child. They might become overprotective and invasive; when that child left the house at 18 to pursue their own life, they would experience a certain amount of sorrow, but more than anything they would suffer if that child were to die.
Sorrow is a complex emotion, but it's an ancient one. For this reason, many people might think that even if desire was indeed the root of sorrow, it would be unavoidable, because desire is so deeply woven into the human consciousness. The Buddha, obviously, thinks otherwise, as his promotion of the art of meditation and the rejection of worldly possessions evinces a belief that we can 'shed' such compromising mentalities. If desire is indeed the root of sorrow, then we must simply stop desiring worldly objects, as these are the ones that lead us to disillusionment and anxiety. The purest desire would undoubtedly be the search for enlightenment: you could not be disillusioned at the attainment of the best possible state of soul, or anxious when Nirvana is everlasting and indelible.
Finally, what of instinctual desires? Basal necessities such as consumption (of food and drink), sleep, or sexual reproduction? The only argument I can imagine for this is that someone on the path to enlightenment and the transcendence of sorrow would simply have to engage in this activities without desiring them. I am overall unclear on this theory, however, and would like to turn it over to Viri or another philosopher to explain.
I'd also like to hear anyone else's thoughts on the prompt Viri has offered.