No, the rendered result is not <img alt="" src="<?php $site->image_02 ?>"> but <img alt="" src="https://mysite.de/media/pages/blog/slug/61586985-1585327526/01_grosse-bilder01.png">. So all you need is to get the complete rendered img tag and the value of the src attribute, and wrap that within your link.
I think I would prefer a custom image tag instead that renders the result differently depending on template though, rather than the hook. Would prevent having to do the regex voodoo.