There's a lot of people over-reacting to this situation. If you read Twitter's blog post its clear that they're following up on enforcing the API "Rules of the Road" they've had for over a year. https://dev.twitter.com/blog/delivering-consistent-twitter-e...
Section !.5. clearly lays out the rules clients must follow "Your Service may be an application or client that provides major components of a Twitter-like end user experience (a "Client"). An example of a Client is a downloadable application that displays user timelines and allows users to create and search for tweets. If so, certain additional terms apply, including:"
These rules are actually pretty generous and plainly lay out their expectations for third party applications. It's unclear what led to Tweets being removed from LinkedIn, my guess would be that Twitter attempted to enforce their branding rules, LinkedIn declined and instead said they would remove the Tweets from being displayed entirely. It doesn't sound like Twitter just cut off access to LinkedIn or came up with new restrictions they haven't already been publishing (and compelling 3rd parties to comply with) for a long time.
Any business must deal with existential threats, be it from competitors, regulation or bad investments. Building a business on top of an API is just another potential risk, you need to decide if they company in question is a safe enough bet to play on, just like you have to decide the same thing for all other risks.