Thank you for your question and for your clear description of your use case. I am a Product Manager at Dropbox for our Sharing Platform team. Our team works on the underlying technology that powers the link sharing you're using. The behavior you describe, and which Hannah confirmed, is correct. For files, the shared links are currently user-specific.
Now, in most cases, as you've said, this works fine. However, we are listening to our users and are always seeking to making Dropbox even better - especially when it comes to the ease and reliability of sharing. In cases where that original user loses access to the content, disbands their account, etc. that original link would break. That isn't a great experience. So we're fixing it!
In the future, all of your shared links will be tied to the content - instead of the user - which will eliminate this scenario. In fact, some content types, like folders, already function this way - give it a try!
I hope that helps and please continue to provide feedback in our community! It's a valuable way for those of us working on our products to keep in direct contact with our users.