Found part of the solution! Thanks to @wjandrea for asking my VS Code version. The official instructions mention the need for VS Code 1.96 or greater and I'm on 1.94.1 currently. This explains the current "correct" method part, but I have yet to see a kosher option to download a .vsix of a previous version, or how to properly download .vsix from browser. It seems to be an intentional removal by Microsoft, although a bit of a strange one.