> the mechanical oscillator that makes the watch advance (it drives the pallet fork and escapement wheel, which produces the ticking sound)
The balance wheel doesn't drive the escape wheel, it's the other way around! The mainspring powers the train, which eventually drives the escape wheel, and the escape wheel provides energy to the balance wheel via the pallet fork.
(If you think about it, it's obvious that it couldn't be the other way around, else the watch would stop very quickly as the balance wheel runs out of energy).
The pallet fork is driven by both the escape wheel and the balance wheel, at different points in the cycle.