Could smartphones benefit from the same charging mechanism as in electronic watches?
Feasible: yes

Useful: no,
because the power needed by a smartphone is orders of magnitude larger than what such a (kinetic) generator generates.

A (non smart) wristwatch needs only very little power, it can run for years on a small button cell. This amount of power can easily be generated by a kinetic generator.

A smartphone has a large battery and needs charging almost daily. Unless you want to violently shake your phone for a couple of hours every day, a kinetic generator will by far not satisfy the power needs of a smartphone.