Summary: LCHATCHILA one can do bedikas chometz on ANY night BEFORE אור לארבעה עשר without a brocha, provided that one room is left for bedika on the eve of 14 with a bracha.
The chiyuv to leave a room for bedikas chometz makes your entire point about being ready to leave before anyone else moot.
Leaving out a detail which changes the entire halacha is indeed distorting halacha.