Trust. I think it has been earned - so far. NCSA Mosaic was my second browser, and while I’ve experimented with all the major browsers and many not-so-major contenders over the interim couple of decades, Firefox has been a constant for me ever since it was first spawned off of Mozilla. I still believe the Firefox team when they say things, and so far that has worked out just fine.
There is no install needed, you can just edit permissions and make the file executable and then when you open it or click it the app runs.
What won’t be created by default is an application menu to run it from whatever desktop environment you use. You can create those if you wish. You can create a launcher in the menu manually, or you can use a tool called AppImageLauncher to create these for you.
There’s a pretty good explanation here: https://askubuntu.com/questions/1311600/add-an-appimage-application-to-the-top-menu-bar