More importantly, you clarify that RZAT is a necessary feature for what I'm mentioning to work properly. You're right. They should both be ensuring the blocks served to customer VMs are zeroed on use and ensure that they are appropriately running TRIM commands to ensure maximum performance from their hardware. Not all SSDs perform RZAT, and it wouldn't be a bad idea for the host to ensure the device is logically zeroed for the VM anyway.
DigitalOcean could easily switch to doing both, or at least guaranteeing the former by creating new logical disks for customers as every other vendor does. If, as they have blogged about in the past, they are directly mapping virtualized disks to the host's LVM volumes, they are unnecessarily complicating their hosting set up and making their host configuration more brittle. With thin-provisioned/sparsely-allocated or with file-based virtual disk images, they can more flexibly deploy VMs with different disk sizes with minimal changes in host configuration.
Alternatively they could trivially ensure that even forensic tools would have a very difficult time erasing volumes by enabling dm-crypt on top of LVM, and resetting the key every time a virtual machine is deleted. This could reduce performance on some SSDs (particularly SandForce based models) but would allow minimal changes to their configuration to ensure deleted data is unrecoverable.