 Affine combinations are independent of the chosen origin

Keep the basis fixed and replace the origin by a point with old coordinates $o$. Each point's coordinates change from $p_i$ to $p_i-o$. For coefficients with sum one, the weighted coordinates becomeAdding the new origin $o$ gives the original point. This proves origin independence rather than asserting that the coordinate tuples themselves stay unchanged.

The requirement cannot be omitted. In the example $A=(0,0)$, $B=(2,0)$, blindly treating $A+B$ as a point gives $(2,0)$ in the old coordinates. If the new origin is $o=(1,1)$, the two coordinates become $(-1,-1)$ and $(1,-1)$; adding them and restoring $o$ instead gives $(1,-1)$. The coefficients sum to two, so this procedure has no origin-independent interpretation as a weighted point.