Is there a way to preserve the links that connect Excel files that are stored on a file share but are moving to SharePoint?
I don't think so. In theory, you could map the SharePoint library as a drive, and then create symlinks to the new file location. That would be quite a bit to configure and manage, and prone to failure with connectivity issues. That solution would probably not work if you are talking about internal links between Excel workbooks/files.
There are 3rd party tools that help update internal references with the new locations. I have not used these, so not sure if they will work in the scenario, but if may be something to check out.
It's worth the upfront effort just to update anything that needs it to point to the new location. If you have applications that need access, and they don't support SharePoint API's, I recommend you leave those on a file share.