A normal copy consists of a program reading from one file and writing to another. There is no way for the filesystem to do a reflink in that case, it just sees that the program is reading and writing stuff. In order to do a reflink, the program must tell the filesystem what data should be “copied” to where using FICLONE or FICLONERANGE. Though some programs will do that by default if possible nowadays when copying files or when moving files between different subvolumes on the same partition, including the Coreutils cp, mv and install commands and some GUI file managers.
One thing that you do still need root for however is proper backups. The built-in backup system is tied to Google, and it’s very limited in what it will back up, nowhere near what you can do with root backup software like Titanium Backup.