is there any advantage mounting an LV by UUID instead of by path?

I know I can specify mounts in fstab by either putting their path (like /dev/sda1 or /dev/mapper/myvg-logicalVolume1) or by fs label (LABEL=root) or by UUID (UUID=1234-5678-...).

I see a clear advantage in terms of reliability using the UUID for "classic" partitions like /dev/sda1, because if you repartition your drive/more partitions around/add more disks it may be that your some of your partitions now it gets recognized with another name, although mounting by UUID is more difficult to tell in which partition/LV your data is being stored.

But using LVM, my guts tell me that the LVM system itself manages the discovery of their disks/partitions and it doesn't matter if some PV is (after playing with partitions/disk) now named different. So there won't be any difference (speaking of reliablity) mounting by UUID or using the path like /dev/mapper/vg-lv, and the latter is more clear.

Is this correct?

That's correct.

Mounting by UUID is one way to work around the old issue of partition names like /dev/sda1 changing if you put another drive in.

device-mapper will persistently name your LVM volumes into /dev/mapper/vg-lv so you can rely on this abstracted name to stay the same, regardless of changes to the underlying storage.

The same goes for devices handled by device-mapper-multipath either without using friendly names (/dev/mapper/WWID) or using friendly names and a bindings file (/dev/mapper/mpath0).

You'd only ever shoot yourself in the foot if you wanted to rename the volume group or logical volume later....(lvrename or vgrename).

Forgetting whatever reason you renamed the vg or lv, the action would screw up your mounts and exports.

LVUUID remains persistent through vg and lv rename commands.

It may be good to use UUID for this reason alone, especially if you are responsible for a large number of exports and mounts.