What commentary on society was John Keats making with an "Ode to a Nightingale"?

I wouldn't interpret this poem as Keats making any particular kind of comment on society as a whole. He is not writing in response to any particular social change. Rather, he is lamenting the vicissitudes of life in general, and is doing so from the point of view of someone, it is widely believed, who was suffering from acute depression. Indeed, Keats's sentiments in this poem draw towards the suicidal: he has "been half in love with easeful Death" for some time, and envies the nightingale, who does not suffer either the speaker's difficulties or his troubled thoughts.
The "drowsy numbness" which pains the speaker and causes his heart to ache is not ascribed to any particular social issue, but rather has arisen because the speaker is so envious of the bird, whose "happy lot" seems so far outside the speaker's experience. The bird, like a "Dryad of the trees," soars, both literally and symbolically, above the speaker's cares.
The speaker yearns for some kind of draught in which the good feelings he half-remembers could be contained: "a beaker full of the warm South" which would allow him to "fade away into the forest dim." He yearns to leave his life and achieve a new existence with the nightingale, free of "the weariness, the fever and the fret" which form such a part of the speaker's own existence.
Keats certainly had much to be concerned about in his own life at the time of writing. He suffered from tuberculosis, an illness which would eventually end his life. His brother also succumbed to tuberculosis—this is alluded to in this poem, when the speaker laments how "youth grows pale, and spectre-thin, and dies." Keats was also unable to marry the love of his life because of insufficient funds to support her. As such, he has reason to interpret life as simply a state in which "men sit and hear each other groan," beset by "palsy" and early death. It is unsurprising that the difficulties of Keats's own existence have forced him into a position where "but to think is to be full of sorrow" and where "there is no light."
In the nightingale, the speaker sees a "light" which is unreachable on earth, or at least of which he cannot conceive in his extremely depressed mindset.