One alternative might be to use iTerm2 and setup a “Quake” menu linked to a hotkey. Then whenever you press something like Alt+' (i.e. the option key plus the single-quote key), a terminal drops down from the top (like in the game Quake) and you can interact with it. Once you’re done, press the hotkey again to dismiss it or just click anywhere outside the terminal window.
Taking this approach, it feels like having “Terminal” embedded in every app because it is just a quick keystroke away.
Disclaimer: YMMV! I’m a software developer so this might seem like an easy option for those who are tech-savvy but not so easy for those who aren’t.