Removing "broken links", or requests that result in 404/410 errors, avoids wasteful requests.
As your website changes over time, it's inevitable that resources will be moved and deleted. If you don't update your frontend code accordingly, the server will issue 404 "Not found" or 410 "Gone" responses. These are wasteful, unnecessary requests that lead to a bad user experience and make your site look unprofessional. And if such requests are for resources that can block subsequent browser processing, such as JS or CSS files, they can virtually "crash" your site.
In the short term, you should scan your site for such links with a link checking tool, such as the crawl errors tool in Google's Webmaster Tools, and fix them. Long term, your application should have a way of updating URL references whenever resources change their location.