Fantastic plugin, though the search functionality is indeed a bit limited. Here's some feature suggestions:
1. different label for the button and the default input value
2. ability to turn off cats/cats descriptions for search result only
3. filter out duplicate results (occurs when a link is in multiple cats)
4. show searchresults from certain cats only (option to use ID's similar to "categories to be displayed" in the general settings)
5. multiple input boxes (or dropdownlists) with a boolean operator between them, allowing a user to find a link that is in cat x AND cat y. Or when already displaying cat x, allow a user to filter on cat y or another search term.
I'm using link library with many links and many are in multiple cats. Problem was duplicate search results and cats/cats description being shown. I noticed that the search results are filtered by the "Categories to be displayed", yet doesn't prevent links from being displayed when clicking a category link (not sure if that's a feature or a bug).
So, what I did was create a category for search results, put its ID in "Categories to be displayed" to make it a search filter and added that category to all links, thus showing all results as usual, but without duplicates. And with the option to display a custom "search results" description. To display all category links on the page I'm using the custom cats shortcode.
I'm telling this because I hope that you will either not remove the search behaviour, or add some of the features above.