Oh, I know the answer. You have to create a project first. Then and only then the other commands will become available for you
Well, yeah. That's the answer. But just to make it cleaner for you. All the commands, that exists in the palette works with scratch orgs, which can't be used outside a project, because of project-scratch-org.json file, that defines the shape of your scratches
I was having this same issue after I created a project and tried to open vscode and log in. The solution I found was going to file -> open folder -> choose the project folder created originally. The options to login option will be available in the command palette.
It was confusing since in the explorer panel the file directory was showing the project files.
Try running sfdx update from VSCode terminal (Even if you have just installed fresh from website). After updating, refresh VSCode by closing and restarting. Your Authorization command should work fine.
I also had this problem in spite of the fact that I was trying to run this command while I was in an sfdx project. What helped me resolve this was uninstalling the CLI and uninstalling the sfdx auth plugin then reinstalling both.
sfdx plugins:uninstall auth
sfdx plugins:install auth
npm uninstall sfdx-cli --global
npm install sfdx-cli --global
After doing this both the command palette and CLI command (sfdx auth:web:login) worked.
Just to add this screenshot for my comment under the best answer.
One last trick is to wait for the extension to run after switching to a project folder. you will see an error message after the extension has run. after that you will have the authorize org option.
This was happening to me as well. I'm not sure what's going on, but Visual Studio Code just seems to be moving slowly in general. I just waited a bit in combination with opening the project folder, and the option eventually showed up.