I can't find the Stashes (nor REMOTES, BRANCHES, FILE HISTORY) menu in the Source Control (the one you open up with Ctrl-Shift-U) tool bar.
This is how my Visual Studio Code looks like:
And this is a picture I found on the web with the menu options I'd like to restore:
I couldn't find an option to restore these menu options.
it seems like somehow GitLens, instead of replacing the Source Control panel, add his own:
You need to install (or enable if you have it disabled) the GitLens extension. That is what adds those sub-sections seen in your second screenshot.
Alternatively, if you click the ellipsis that is to the right of each of your repositories, you should see a "Stash" menu item that should also meet most of your needs.
Note that the GitLens extension is apart of the Git Extension Pack extension, which contains a few other extensions as well (including Git History, which you appear to have already).