The problem is when I click to add a search input nothing happens (no search input appears). The same thing happens if I try to add a search/reset button as well.
Thank you for the screenshot and additional details.
Given that I don't have access to the site then I can't fully debug the issue. Are you able to record a video of you trying to add the search so that I can see what happens on the backend.
Unfortunately it would appear that your theme is causing some Javascript errors in the console which causes the view setup to fail.
You should be able to work around this by using the classic view setup by going to Toolset->Views and building your view this way. However the issue is definitely with your theme as the other 2 themes that you have available on your site works fine.