Some issues could be prevented if push messages added end-to-end encryption by default, something that shouldn’t be particularly hard to use if it was built into the dev tooling. Instead, developer recommendations like this one [0] suggest that you should put content into your push messages and optionally use a separate library to encrypt them. Clearly developers aren’t doing this, hence the opportunity for surveilla…
The timing would still give you away - with a privileged network position you can tell that a user sent a message to an messaging service, and that some set of users got notifications from that messaging service moments later. Observe that enough times and you'll have good confidence in the members of a group. If you're trying to hide from that type of attack you need to send a fixed rate stream of messages (most of…
It's all about the timing (and meta-data like which app), not about the contents.