I am not sure if "why" is a good question. Anyway, there are many phrases using the genetive singular in a similar manner:
I daresay they all describe the "mode" of doing something.
For me as a very oldfashioned native speaker it is absolutely natural to somewhat freely form phrases of this kind:
Zornigen Blickes nahm er seinen Hut und verliess das Zimmer.
But the number of combinations that sound native is very limited. I would never say:
Schweren Armes hob er sein Schwert.
It would be good to know if this ever was standard practise. I tend to assume that.