Removing search icon from the header

Agreed.

Technically users could go into the dev tools, find the rule, and uncheck the display none and then use the Discourse search.

But I seriously doubt many would. In fact I think if you used a less obvious selector that the owner would never become aware of it. eg.

div.panel ul.icons li:first-child {
 display: none;
} 

I don’t understand the request, IMHO Discourse search works great. I’m guessing the owner has been listening to some SEO snake oil.