Unfortunately I cannot access KP's link from home and I don't know Cantor's original argument. However, my favourite argument for $|A|=|\mathbb R|$ is as follows (your non-injective argument actually, just looked at more carefully):

Every number $0.x_1x_2\dots\in A$ gets mapped to $\Sum_{n=1}^\infty x_n2^{-n}$, i.e., we consider $0.x_1x_2\dots$ as the binary representation of a number. This map is not 1-1. However, it fails to be 1-1 on only countably many places, namely, a number $0.x_1\dots x_n0\overline 1$ is mapped to the same real number as $0.x_1\dots x_n1\overline 0$. But there are only countably many pairs like that.
So, your map fails to be 1-1, but only at countably many places, and at each failure of injectivity, only two numbers are identified.

Hence, after removing countably many points from $A$, your suggested map embeds the rest of $A$ into $\mathbb R$ in a 1-1 way. The countable set of exceptions can be mapped outside the unit interval, in a 1-1 way.

Unfortunately I cannot access KP's link from home and I don't know Cantor's original argument. However, my favourite argument for $|A|=|\mathbb R|$ is as follows (your non-injective argument actually, just looked at more carefully):

Every number $0.x_1x_2\dots\in A$ gets mapped to $\sum_{n=1}^\infty x_n2^{-n}$, i.e., we consider $0.x_1x_2\dots$ as the binary representation of a number. This map is not 1-1. However, it fails to be 1-1 on only countably many places, namely, a number $0.x_1\dots x_n0\overline 1$ is mapped to the same real number as $0.x_1\dots x_n1\overline 0$. But there are only countably many pairs like that.
So, your map fails to be 1-1, but only at countably many places, and at each failure of injectivity, only two numbers are identified.

Hence, after removing countably many points from $A$, your suggested map embeds the rest of $A$ into $\mathbb R$ in a 1-1 way. The countable set of exceptions can be mapped outside the unit interval, in a 1-1 way.