I believe the issue is not with you or Sparkle, but with LinkedIn.
I tried to integrate it myself. The badge doesn’t display on my website created with Sparkle, nor on a non-Sparkle website. Moreover, it doesn’t even show up in the preview on LinkedIn’s website for me.
Thanks. I instead use a linked in png logo and then set the LinkedIn page as the destination when clicked .
However wuth sparkle Seo check, the linkedin link is broken but it does actually work on the site