Is it not recommendable to stick with plugins that are no longer supported by the author for production sites?

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As for me key questions here are:

  1. Is plugin outdated? Not using approaches and coding practices of current WP versions.
  2. When will it become outdated? This can range from months (next WP version changes something relevant) to years (plugin does something simple on using common and basic hooks).
  3. Can you handle when it will become outdated? Maybe you can maintain it yourself. Maybe you have budget to hire someone for that. Maybe it will ruin your site.