You need to separate concerns here. Syncing files from OLD to NEW is done by using e.g. rsync or rclone (I’d use the former), but this is a separate thing from backing up your data.
Are you intending to use both computers actively and change the same set of files on both of them? If so, are you really looking for ways to keep two machines using and changing the same files in sync, so that when you change files on NEW the changes are reflected on OLD? If so you need something like Unison or Syncthing - using a backup repository for this is not the right approach.
Restic is something you use to back up your data, not sync your data actively as it’s changing. REST-server is something you use to store the backups of your data. But neither is designed for the use case of syncing data.
Please answer the above, that will give us better information and enable us to give more relevant answers.