Why do you think or doubt it may not work? The document explains how to set up LVM on a new or unused device and how to copy your current root partition. It does not show how to re-use or convert the existing root device, hence I think the title is somewhat misleading and should rather say copying instead of converting.
Perhaps you may also find the following useful, which covers many of the tasks required to perform an in-place replacement:
I would rather opt for BTRFS than LVM. Before BTRFS, LVM was a convenient and easy way to add new disks in order to increase volume capacity, however, considering today's storage standards there is no real need to be stingy regarding disk space, at least not or the root or system partitions.