Safari on iOS does not expose the Vibration API, which is why that button does nothing on every speaker cleaning site.
iOS specific behaviour
iPhones route audio differently depending on whether the microphone is active, and Safari cannot access the vibration motor at all. Both affect how cleaning tools behave.
Running a cycle safely
Turn the ringer switch off silent, set volume to maximum, remove the case, and hold the phone with the bottom speaker pointing at the floor.
Try the free speaker cleaning tool and measure the result before and after.