Not sure about what carrier you use or what device you use, but I'm on AT&T using a Pixel 6a with CalyxOS, which is just de-googled Android, and on Android if you accidentally swipe away an amber alert notification you can find it again by going to:

Settings -> Safety & emergency -> Wireless emergency alerts -> Emergency alert history

But your point still stands, governments and public institutions really need to stop relying on privately owned and operated social media platforms for posting stuff like this. If they want to use a social platform to publish alerts, they would be much better off standing up their own Mastodon instance that is "just" for those alerts. People could follow those accounts if they want, and those institutions wouldn't be subject to the whims of overpaid unpredictable man children.