We are currently using pg_upgrade to go from 9.6 to 13 in our systems. its supported and works well. We are using hte 'hardlinks' feature that makes it extremely fast as well.
Do be aware that using the "hardlinks" feature will mean that restarting the old database version will not work properly and may not be safe. You should make sure to copy the old directory beforehand if you might need that, or simply restore from backup.
I'd argue you shouldn't be performing an upgrade at all without a proper backup. But yes, absolutely do not use a hardlinks upgrade unless you have a backup laying around.
