If you don't want the default catalog search module, then the best way would be disable that module. For this go to System > Configuration > Advanced. This page will show all the activated modules there. You need to disable the module Mage_CatalogSearch module. When you need, you can also enable it through this place. That's it.
But before disabling that module, you need to ensure that, the external plugin that you are relying on does not inherits from default catalogsearch module(Mage_CatalogSearch).
EDIT
As Benmarks pointed out in the comment, if you really want to completely disable default search, then what you need to do is,
Go and find app\etc\modules\Mage_All.xml
Find this section
<Mage_CatalogSearch>
<active>true</active>
<codePool>core</codePool>
<depends>
<Mage_Catalog/>
</depends>
</Mage_CatalogSearch>
Put false inside <active /> node.
Now clear your cache and then you are good to go.
System > Configuration > Advanced > Disable Module Outputdoes not disable the module. It merely prevents blocks belonging to that module from rendering anything other than an empty string. The full block lifecycle up through rendering is triggered. In the case that the<layout>node includes amoduleattribute, layout updates may also be ignored (not the case for CatalogSearch). – benmarks Sep 28 '15 at 08:58