The problem: after “Check for updates…” and “Update now”, VS Code for macOS never restart or actually update (after it is manually restarted). This problem was reproduced on macOS 10.15.4 and VS Code 1.41.0

The solution: run

$ sudo rm -Rf ~/Library/Caches/com.microsoft.VSCode.ShipIt

Reference: https://github.com/microsoft/vscode/issues/77949