Answers
Do web apps work without an internet connection?
Partly. A web app with a service worker can cache its own files and open without a connection, but a messenger needs the network to send or receive — what works offline is the shell, not the conversation.
This is the honest limit of an installable web app. It can start, show what it already had, and queue what you type. It cannot deliver it.
The same is true of a native messenger, which is often forgotten: the app opens, and nothing moves until there is a connection.
What differs is that a native app is generally better at using a brief connection in the background. A web app has to be open, or installed to the home screen with notifications on.