Did you try creating a custom post type, and adding custom fields with Advanced Custom Fields? Advanced Custom Fields makes it easily to add custom fields to a post, and gives the possibility to hide other elements, like the content editor and featured image.
In my opinion, you should create a custom post type, and add a custom taxonomy (tags) to it. After that, you create a custom field for uploading the file. In this way you can set a post title for each PDF-file, which most likely will improve your SEO score. In Advanced Custom Fields you can set the extensions that are allowed in the file upload.
For searching and indexing a custom post type with custom fields, I recommend using SearchWP. SearchWP has the ability to index and search through PDF’s, so the content of the PDFs will also be indexed. It also gives you the possibility to set weight to custom fields, and create supplemental search engines to only show the results in PDFs.