'... a pair of ragged claws' is not synecdoche. Eliot means exactly what his metaphor says, and he doesn't need endless unimaginative critics second-guessing him and thinking, rather ludicrously, that really he means a crab. He says 'claws' because he means claws. He means disembodied claws, grasping, only able to grasp, unable to engage further, freed from the responsibility of engagement, lost in a silent world, picking over morsels, ideas, abstractions, detached from the world of people coming and going and judging, just pure apprehension freed from anxiety, freed from the slow death and banality of rooms and functions and society and coffee spoons. And yet infinitely sad in his loss of it.
And yet not sad, because claws by themselves cannot be sad. The sadness is Eliot's projection onto that abstract world, and expresses the impossible dichotomy of at once being wholly disembodied and free, yet still—from without—knowing the loss inherent in such a state.
A partial allusion in 'ragged claws' is to the compasses or dividers in Blake's watercolour of Isaac Newton. They are also claws, and they represent again this detachment from the outer world. Newton's focus is entirely upon his realm of signifiers, perceiving through his 'claws,' oblivious to the silent, inhuman submarine-scape that now surrounds him and isolates him, in consequence of such determined abstraction.
Eliot at once embraces such a possibility, yet still shrinks from it, as does Blake, whose painting foregrounds the unnatural state required for Newton's fixation. Both of them regard it as something near to oblivion. Blackness with only one tiny chink of light permitted to enter, like the room in which Newton performed his experiments with prisms and refraction. One tiny bead of light, but such brilliant light to force the moment to its crisis, to admit the drama and urgency of a new level of human understanding... But oh, what darkness surrounding it... What sort of life is that?
There may be no such brilliance in the rooms, the tea, the ices, and the deathly, ticking coffee spoons, but they are the stuff of human life and—perhaps unlike the inhuman Newton—Eliot knows he cannot, ultimately, abandon them for the 'floors of silent seas.' His moment has passed; he was too fearful, and the eternal footman knows it. And consequently snickers.
(Hermeneutics: actually, there are three possible places in this complex: the claws, the crisis, and the rooms. He wants to force the crisis, but is intimidated by the rooms. He thinks resignedly, wistfully, of the claws, relinquishes the crisis, accepts the rooms. The claws will remain as latent potency and denial in his secondary levels of expression in the rooms, never to be realised, but a source of intellectual/emotional wishful thinking/refuge.)