To add to Marks Suggestion, you can remove the tag
$:/tags/PageControl from $:/core/ui/Buttons/advanced-search to remove it from the PageControls Menu
and edit
$:/core/ui/SideBarSegments/search and remove the following to remove it from the sidebar
<$button to="$:/AdvancedSearch" tooltip={{$:/language/Buttons/AdvancedSearch/Hint}} aria-label={{$:/language/Buttons/AdvancedSearch/Caption}} class="tc-btn-invisible">
{{$:/core/images/advanced-search-button}}
</$button>
Someone in the know may be able to find $:/AdvancedSearch but it would defeat many attempts, at least casual attempts.
Regards
Tones