Why doesn't every solar eruption bring bright auroras to Earth?

Answer: It must head our way

It must head our way ✓ — Correct! A solar eruption only becomes a strong aurora risk for Earth if it is directed toward Earth and has enough speed and energy. Many eruptions go off to the side or weaken before arriving. So forecasters do not just ask “Did the Sun erupt?”—they ask “Did it erupt in our direction, and how strong is it?”

Earth must cool first — Wrong. Earth's surface temperature has little to do with whether solar material reaches the magnetosphere. A freezing night does not guarantee auroras, and a milder night does not prevent them.

It must be a new moon — Wrong. A new moon can make the sky darker and improve viewing, but it does not control whether Earth is actually struck by solar material. It changes visibility, not the space-weather event itself.