I had multiple sites (in WordPress multisite) where the URL preview wasn't working on LinkedIn: I was only seeing the blog's domain name as the title and URL. Previews were working fine on Facebook and other social platforms.
After days of trying to pinpoint the issue after migrating some blogs and changing my WordPress security plugin to Better WP Security, I finally found that LinkedIn was blocked and Apache was returning an HTTP 403 error.
After investigating, I noticed it is due to activating the "Enable Default Banned List" feature in the "Ban" section of the Better WP Security plugin.
To deactivate this feature, it isn't too intuitive: you need to uncheck the checkbox and click "Add Host and Agent Blacklist". Otherwise, clicking "Save changes" at the bottom of this page doesn't do anything.
I hope this will save days of hair pulling to others.