Using 1:1 mappings of LVM logical volumes to guest VM block devices is the most straightforward and performant method of doing it on Linux, short of doing 1:1 mappings of entire disks or disk partitions to guest VM block devices. While using file-based disk images would prevent data leaks between customers without any further effort required on the VM provider's part (assuming they don't reuse disk images between customers!), there are tons of downsides to file-based disk images, mostly related to performance and write amplification.
I don't agree that file-based disk images are more flexible than LVM's logical volumes — it's ridiculously easy to create, destroy, resize, and snapshot LVs.
Until very recently there were serious problems with putting LVM under any sort of concurrent load. Making more than a few snapshots at the same time, for instance, was asking for trouble. I say "was" - I've got no idea if these problems were fixed. You just don't have those problems with file-based images.
I don't agree that file-based disk images are more flexible than LVM's logical volumes — it's ridiculously easy to create, destroy, resize, and snapshot LVs.