Random is bad because you need some random distribution and by gathering statistics server can determine if your distribution of delays match. Delay until really clicking the button would be highly individual. That is why I said "when they open the application".
I want to address you last concern. moxie's specific concern is that without the extra measure I explained in the blog article, WhatsApp could find out on a large scale, covertly (!!!), who has this setting enabled or not. Simply because after a while pretty much every client will have to face the decision whether or not to retransmit for a benign reason.
Of course if they MITM, they can distinguish the re-send text from the garbage text, but the point is that then they are MITMing already and risk being detected. So instead they could've just MITM the conversation from the beginning on with risking of being detected.
If notifications/blocking disabled (newbie setting):
Send re-keyed ciphertext immediately.
Random time later send garbage (automatically discarded by client)
If blocking enabled:
Send re-keyed garbage immediately.
When consumer notices the popup some (random) time later,
- and clicks "re-send": send re-keyed ciphertext
- and clicks "discard": send re-keyed different garbage.
However, note that if a compromised server MITM, they will probably be able to tell the difference between garbage and actual message (because the server provides the bad key, so can decrypt the immediate response message). It's really not trivial. Don't roll your own crypto... :-)