I am a little confused by your wording. Tab menu (CTRL+M) doesn't really let you "do things that aren't already shortcuts", it just shows your tabs (which I think has been replaced by "Search tabs" CTRL/ALT+SPACE now).
To access the main menu in the menu bar, you can use the shortcuts inherent to Mac, as "main menu inside app" is more of a Windows element that doesn't apply to Mac. If the relevant device shortcuts are already active, then you can use CTRL+F2 (FN+CTRL+F2 without function lock) to focus the keyboard on the menu bar.
Details here: https://www.intego.com/mac-security-blog/navigate-the-menu-bar-dock-and-more-using-your-macs-keyboard/