Every service has "terms of service" which are the conditions that you are allowed to access the service and what you may do with the service once being granted access. For example, if you start pouring toxic waste into your sewer, you will find that the city will both disconnect you from the service and they will fine you for violating the terms of service you nominally agreed to when being hooked up.
In LinkedIn's case, they allow you to access their service, with HTTP, to render a page in a browser for viewing of that page. Full stop. Any other use of the data you acquire over HTTP, or any other method of acquiring said data over HTTP is disallowed by the terms of service.
Not only does LinkedIn have a legal right to stop scraping after the fact, they have literally centuries of common law in support of that position.