Go to Nexus Mods and find, "Intel HD Graphics Bypass Package," under Fallout 3. While it says Intel Graphics, this also works for AMD.
Once you've downloaded it, navigate to your Fallout 3 local files (Found in properties after right clicking Fallout 3 in your library) and move the d3d9.dll file from the bypass package into your Fallout folder. After that, go to documents>my games>Fallout 3, and then drag the Fallout document into that folder, and replace the old Fallout document.

This is one of the most common fixes for Fallout 3 crashing after the launcher.

Go to your Fallout 3 local files, right click the Fallout 3 executable file, go to properties, and under the compatibility tab, run the game with the Windows XP compatibility mode.

Follow the steps in method 2, except this time instead of running the game in compatibility mode, tick the run as administrator box.

In file explorer, search for falloutprefs.ini, right click on it, and choose open file location.
After finding it, open it in notepad and find bUseThreadedAI=0 and change the 0 to a 1. If bUseThreadedAI=0 is not there, then just add bUseThreadedAI=1 yourself.

Find Fallout Anniversary Patcher on Nexus Mods and download it.
To install it, unpack the contents of the archive to Fallout 3 folder and run Patcher.exe. After doing this a black console box should show up which will tell you if there was any error during the installation. Once it's done, you can run Fallout 3, but don't use FOSE loader to run it.
You can also find a video guide for this installation here

Those were all of the methods necessary after the 2021 patch. If you look for methods online, you may see a method telling you to install games for windows live, while this would've been required a few years ago, after the 2021 patch the dependency on gfwl was patched out.

And as a final suggestion, as with any game, always make sure your driver is up to date.