The dominant attitude of all of Thomas Hardy's work, both prose and poetry, is pessimism. To ask why the speaker in "The Darkling Thrush" does not feel the joy the bird does is basically to ask why Hardy himself focused relentlessly in his writings on humanity's pain and suffering.
A lyric poem such as this is like a snapshot which is complete in itself but does not provide the explanations and elaborations we find in a novel, for instance, or often even in a short story. Think of the meaning of the lines:
So little cause for carolings
Of such ecstatic sound
Was written on terrestrial things
Afar or nigh around . . .
"Terrestrial things"—the conditions of mankind on earth—are always seen by Hardy as dominated by randomness and sorrow. To understand his viewpoint, presented in "The Darkling Thrush" (and other poems such as "Hap" and "Channel Firing," to name just two), I would suggest reading some of his novels, if you have not already. In Tess of the D'Urbervilles, for instance, both chance and the unfair attitudes of society contribute to victimize a young girl and to destroy her, in spite of her innate goodness and innocence. In The Mayor of Casterbridge, a man is destroyed by his own past mistakes and past cruelty in spite of his efforts to reform himself. Hardy's poems are compressed versions of the bleak view of life expressed in much greater detail in his novels.