As a web developer, I do a thorough scan of all links on all pages of any website I am developing as soon as it is complete. There is software such as Xenu’s link slueth which goes page by page link by link. You specify how many levels deep to scan. The end result is a long list of links and the status of all those links. I do this every for every single website, multiple times. Here is why:
The impact of broken links on your website a twofold. Firstly, there are search engines. Search engines cannot index any pages if the links are broken. In addition any authority will not be passed onto those pages and that will affect your ranking in the search engines. The second is that visitors who find broken links are likely to leave your website. Most visitors have very little patience and will move on or go back to the search they came in with.
Many website developers use a CMS such as WordPress to design their websites. This can involve third party themes and plugins. While a lot of these can be innocent, there are known cases of plugins inserting links into pages. You will not know these links are there. From this they can either pass on page authority to sites of their choosing or potentially install malware to your visitors. If you use free plugins and themes, always scan your website after development is complete.