Yes indexing is needed for AI to better understand your codebase and so it can lookup parts, however over remote connection that can lead to slowdown specially after upgrade to a new version as it re-indexes the code.
Sometimes disabling and re-enabling indexing may help. At least it would show if this was directly causing the issue.
Is the project large? or has large files that need to be fetched from remote server?
Additional option:
Cursor released with version 0.50 a new Remote SSH extension, please try replacing the original with this one. See instructions on following page provided by Cursor team.