Search in relational/display fields

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The issue
Page and Area - relational fields (separate tables), visualized through Display template.
Tag - the only field that directly belongs to table Contents.
Text - relational field/translation, displayed through Display template,

The issue is that I can only Search on Tag field that directly belongs to the Contents table (I can find "aboutus"). If I want to search in Page, Area and Text fields, I have to setup custom filters (I can't find "about us" or "footer" with standard Search). It would require 4 separate filters to search in all displayed columns which is not so convenient.

Idea
Could the "Search in items..." work in all relational/displayed columns? Either by default or even better with a possibility to select searchable fields in collection data model.