Nobody is going to know for sure if its a false alarm, or if we'll wind up with a murdered child until after the fact, so why wait or ignore it?
As for the risk of a murdered child, well, if that argument works for you, then why stop there? Why should police have to wait for a search warrant...what if we end up with a murdered child? Why should police have to avoid using deadly force to stop a fleeing suspect...what if we end up with a murdered child? Of course, we already have a (rather large, vocal) group of people who believe exactly this, which is why CPS is called on parents who, for example, run into a bakery, with their minivan idling with 6 kids in it, and get arrested for child endangerment.
It's easier to take an extreme, confident position when you don't know anything about it.
The idea doesn't scale unless the events are actually quite selective, and a missing person that isn't even necessarily missing yet probably shouldn't qualify.
Even if you are sympathetic and care about that kid, it doesn't matter because it doesn't work anyway. Everyone just ignores the alert. It's just not a reasonable balance between how much some bad thing matters and how many people's attention are commandeered for it. Every single death even from peaceful old age after a rich full life is a tragedy, but the rest of the world can not care about it. Literally can not, because it doesn't scale. Those tragedies are happening at a constant rate of many new ones every second of every day at all times. Instead there are a much smaller set of people who care about or who's job is to care about each one, and that set is smaller than "everyone in a 400 mile circle". The fact that a person can drive 200 miles in any direction in a few hours doesn't change that.
It's probably a good thing for the system to exist, but if it's used the wrong way, it's entire utility and reason for existing is sabotaged and nullified.
However the real point not about the alert system but about a carrier's ability to control your device more than yourself, even if you own your device outright, that I do not say is a good thing that it exists. I accept that there's not much to do about it. I do have a rooted phone running LineageOS, which gives me a bit more control, but I don't kid myself that that really means much. But I don't think it's good or right.
If you don't, we'll end up with a murdered child, so why not?
No one should presume they have an IRQ line to your brain and attention. They don't get to interrupt you with bullshit you couldn't care less about.
My phone blaring at me at any time w/o warning and with no way to control this behavior seems quite a bit more intrusive to me than your examples.
Amber/Silver/Blue alerts should be more like a regular notification.
https://www.tandfonline.com/doi/abs/10.1080/0735648X.2014.10...
Once I found out that I could disable them, I turned them off entirely.
I'd be more okay with them if they didn't all use that horrible klaxon noise. That noise usually indicates an immediate threat to life, which doesn't feel appropriate for Amber or Silver (which indicate a potential threat to a particular person's life in a rather large area). Save the klaxon for "we're getting nuked" and tornadoes.