Mouse recorder for Chrome
The click is a point
The recorder keeps the place you click. It is a point on the page, not the name of a button. On playback the click returns to that point.
Mouse movement is a choice
Before the first recording you can include how the mouse moves. Once a scenario is saved, that switch is hidden. Record again and the saved choice is used. Clicks are kept either way.
Play it where you started
Press Play on the tab where you want those clicks. Speed is half, normal, twice, or four times. If the site ignores anything except a real mouse press, the click may not count.