As the crow flies isn't archaic.
The matter can benefit a bit from some context, though it doesn't change the answer at all. As the crow flies means without encumbrances/obstacles (as a land animal might encounter.) The preceding chapter ended with this critique of a London court:
If all the injustice it has committed and all the misery it has caused could only be locked up with it, and the whole burnt away in a great funeral pyre — why, so much the better for other parties than the parties in Jarndyce and Jarndyce!
The whole book is basically about the injustices committed by the legal system in England.
The next chapter changes scenes from the Courts to "Fashion"
It is but a glimpse of the world of fashion that we want on this same miry afternoon. It is not so unlike the Court of Chancery but that we may pass from the one scene to the other, as the crow flies. Both the world of fashion and the Court of Chancery are things of precedent and usage; oversleeping Rip Van Winkles, who have played at strange games through a deal of thundery weather; sleeping beauties, whom the knight will wake one day, when all the stopped spits in the kitchen shall begin to turn prodigiously!
Dickens is saying, it doesn't require much of a change in the way one is thinking - it's not much of an obstacle, but in fact (wittily, as Dickens writes) pretty straightforward - to jump directly from the Courts to Fashion, they have so much in common.
As the crow flies is recorded from 1767 (an earlier usage is less clear), and the meaning has not changed; it means 'directly, in a straight line'
The Spaniaad [sic], if on foot, always travels as the crow flies, which the openness and dryness of the country permits; neither rivers nor the steepest mountains stop his course, he swims over the one and scales the other.
It has a French equivalent, à vol d’oiseau.
But that is archaic, but still encountered often enough.