When executing LinkedIn tasks in Outreach, you're not seeing the LinkedIn UI element in the task panel. Instead, you have to clicking on the LinkedIn link and go directly to the LinkedIn site.
Executing LinkedIn tasks in Outreach
LinkedIn Sales Navigator accounts
While working on the task, click on the cogwheel in the top right of the task panel. Enable the LinkedIn Sales Navigator option, and click Save.
You can also enable this in your user's account settings page as well.
If you do not have a LinkedIn Sales Navigator account, and/or this option is disabled, it will not show the LinkedIn UI element directly within your task panel. Instead, you will have to go directly to the prospect's LinkedIn page.