This is in no way specific to delivery reports. It applies to responses to requests via connections of any kind, in general.
random/varied app notification delay
Uniform random delays can be averaged out. It's not really a good way to mitigate a timing side channel attack. It doesn't work for cryptography. In that space, you have to make operations take constant time, not introduce random delays.
but a system-wide random/varied delay should help
The OS doesn't have a way to provide a universal delay for non-OS-based messaging systems. There is not some special OS API for sending delivery notifications in arbitrary messaging systems. That's not how messaging software works. It's also in no way specific to this kind of request and response.
It sounds like what you're requiring is adding random latency to connections. You can do this via a VPN app and we don't think it belongs in the OS, particularly since it's not really going to accomplish the goal.