I just did a factory reset on my device and "Drawing over other apps" is enabled for over 170 apps. I read that only a very short list of apps need this permission. Do 170 apps really need this permission by default?
No, it is not normal. Usually you need to give permission depending on the android version you have (Beginning in Android 6.0 (API level 23), users grant permissions to apps while the app is running, not when they install the app).
Very few apps need the draw over other apps (for example facebook messenger with the chat heads, and other apps that put an overlay on your screen).