In a Socratic vein, this answer does not have a lot to add to what you have already said yourself. Goddesses of the hearth, such Hestia and the more civically prominent Vesta of Rome, are not only divine centers of the family household and city, the primal "homebodies." They are also associated with earth, to most Greeks the stationary substance, and being itself.
Hestia is rarely portrayed in myths and in effect "doesn't go anywhere," not even in linguistic conveyances. Stationary to the core. So it is only natural as a literary device that Plato would have to make this exception in an event where "all the gods go off somewhere," so to speak. Their upward and outward advance into knowledge of the Forms is figuratively balanced against the fixed, earthbound center.
It may be that Socrates is simply pointing out the obvious qualifier, that in talk of a "procession of the gods" this one goddess wouldn't join in, of course. Goes without saying. Certainly, the dialogues are rife with young men eager to point out inaccuracies, "but what about..." or "that can't be right..." So merely a preemptive note to quiet the attending puppy.
Or it may have a broader literary function. It may be that Plato's dialectical imagination is picturing here a great unfolding, a division of earth and heaven, as when a movie necessarily indicates a rocket heading into space by picturing it in relation to a stationary ground below. Our eye is with the rocket and its adventuring crew by means of contrast with the receding earth.
And just imagine if Hestia had gone off with everybody else! Can we picture the essence of centeredness hopping into a chariot? Who's watching the homefront? How do we know where we are? The one thing Plato can't bear is things tipping over into relativism. Uncentered boundlessness, the vice of Sophists!
We recall too, that Socrates is making a rare perambulation outside of the city walls, drawn out into the tipsy realms of love. And he was such a determined homebody that he was perfectly willing to die just to stay put. There may be a touch of rectitude, a mannerism of propriety in his nod to the hearth keeper.
Of course, this is all pure speculation. One could have (and did have) fun weaving analogies and hypotheses. But I think you were right to begin with and there isn't any deeper meaning or significant negation here. After all, to be seeking a deeper meaning by looking back at solidly grounded Hestia would be, in Platonic parlance, seaching in the wrong direction.