Click-activated animations do not work when I publish Presenter

That isn't how the animation cues translate in Presenter.

The timing of the animations is based on the audio, or silence, on each slide. There is 5 seconds of silence on each slide, but you can add or remove silence in the Audio Edit tool.

On-click animations can be placed wherever they are needed to cue in the presentation. By default they are all placed at the end of the slides audio/silence. This can be done by moving the click markers in the Audio Edit tool or by using the Sync tool to cue the animations when desired.

If you want the user to have to click to move on after each animation, you can turn on the 'Pause after each animation' feature under Presentation Settings > Playback > Pause after each animation. The viewer will have to click the Play button to move on, not on the slide though.