Setting List Grid as Default for Product Display Using Search Bar

I'm not sure I understand your question correctly.

You want to show categories which match the search query, correct?

I would implement a new block, add it via layout.xml to the catalogsearch_result_index and/or catalogsearch_advanced_result.

The block only needs to get all the categories which match the query, here you need to define whether you want like or fulltext search, or something more complex.

Then you need a template, iterate over all the categories and output them.

To set default list mode for search result page, you can copy catalogsearch_result_index.xml

from:

vendor\magento\module-catalog-search\view\frontend\layout 

to:

app\design\frontend\Your_Vendor_Name\Your_Theme_Name\Magento_CatalogSearch\layout

add this code:

<arguments>
    <argument name="_current_grid_mode" xsi:type="string">list</argument>
</arguments>

into block name: product_list_toolbar