The bad consequences are diffuse, abstract and distant (conspiracy-looking, tinfoil-like), while it's very easy to viscerally understand that "even if they just save one child, it's already worth it".

They should give precise numbers of how many such crimes are detected via such means or are expected to be detected per year, and how many of those are not possible to catch through regular investigative work. It just seems ridiculously out of proportion especially that with all this flurry around the topic, the criminals surely aren't using WhatsApp for this any more, but especially won't be once the law is adopted. Sure, many are likely stupid but if they are so stupid, won't they fall into other honeypots?

Why are chat apps the best leverage for uncovering this? They'd have to justify this with some sort of data and numbers.

Because later they can just come back and say, well unfortunately they are now all using other means, so now we need to break https,we need to ban e2e, we need to ban vpns, tor and foss operating systems etc etc.