Whose website? Does my local city have one? Do they have the IT resources to ensure it is hosted? Do they know how to get a .gov domain? What about state news? Lot of friction and multiple potential locations vs a Federally run resource.

Even if you don't use it, you can check it during an emergency. Even if you don't check it, someone else else was checking it and relaying that information to you. You used to be able to search an account on an app you already have, and get the latest information. Now governments have to manage their own instances, you have to search their website, find the url, subscribe to it or download a custom app, and then you can finally see what used to be a search away from you.

That can only be described as a loss.

Yeah. Mobile phone alerts are probably about the best you can do these days beyond going house to house (and even then). You're never going to get to 100%. But, as you suggest, Twitter isn't even a particularly good channel. I use Twitter and I have no idea what my local government channels are, if any.