Does the SQL Server Service Account have to have access to a network share to perform a backup?
If you're performing native SQL Server backups, the service account running the SQL Server database engine must have access to the backup location. The Agent job can run under a proxy to fire off the job itself, but the engine still performs the backup so still needs access.

The only way to have the backup write to disk as another user would be to use a third-party tool that receives the backup data through the SQL VDI interface and writes it to disk as the appropriate user.

From Docs:


  Ownership and permission problems on the backup device's physical file
  can interfere with a backup operation. SQL Server must be able to read
  and write to the device; the account under which the SQL Server
  service runs must have write permissions.


For managing the files after they've been backed up, you can use a SQL Server Agent proxy for this to ensure you're using the correct account for file system access.