Possible to use tags/buttons as exposed filters?
It sounds like you want to have something like this, what is provided by the Facet API module (Drupal 7) or Facets For Drupal 8.


Image copied from: https://www.webwash.net/drupal/tutorials/intro-search-api-part-2-faceted-search (explanation of Facet API combined with Search API).  It should however also work with the Drupal core search system. (I only used it with Search API)


  The Facet API module allows site builders to easily create and manage
  faceted search interfaces. In addition to the UI components that come
  out of the box, themers and module developers can build their own
  widgets that can optionally be contributed back to Drupal.org. Facet
  API works with the core Search, Search API, and Apache Solr Search
  Integration modules (including Acquia Search) meaning that code and
  configuration can be reused as-is with the most popular search
  solutions available to Drupal. Refer to the documentation for more
  details.


Each facet will be exposed as a block, that you can place anywhere in your site from the block administration page.