@Patent : "sorry, i wasnt exact enough: 'click' also fires, but it is doing other things!"
No, no, ...When click() is used with an element it fires the element's click event. an do nothing else like firing "mousedown" as you indicated.
the problem is that in your question you talk about MouseClick (which is a javascript event), while you should have talked about the action : "click on a button
Clicking a button generates 3 events:
The code of the page you are using, seems to have two event handlers installed: mousedown, and click. (also check if your page does not handle mouseup)
So to simulate the action of clicking on the button you have to simulate the two events : mousedown, and click
the code you posted with Jquery simulates these 2 events. It is better in 2022 to remove the use of jquery and to do this in pure javascript.
for example pure javascript code
let elt=document.getElementsByClassName("myButton")[0];
elt.dispatchEvent(new Event("mousedown"));
elt.dispatchEvent(new Event("click"));