education - What mistakes, if any, were made in Numberphile's proof that $1+2+3+cdots=-1/12$?

This is not a duplicate question because I am looking for an explanation directed to a general audience as to the mistakes (if any) in Numberphile's proof (reproduced below). (Numberphile is a YouTube channel devoted to pop-math and this particular video has garnered over 3m views.)

By general audience, I mean the same sort of audience as the millions who watch Numberphile. (Which would mean, ideally, making little or no mention of things that a general audience will never have heard of - e.g. Riemann zeta functions, analytic continuations, Casimir forces; and avoiding tactics like appealing to the fact that physicists and other clever people use it in string theory, so therefore it must be correct.)

Numberphile's Proof.

The proof proceeds by evaluating each of the following:

$S_1 = 1 - 1 + 1 - 1 + 1 - 1 + \ldots$

$S_2 = 1 - 2 + 3 - 4 + \ldots $

$S = 1 + 2 + 3 + 4 + \ldots $

"Now the first one is really easy to evaluate ... You stop this at any point. If you stop it at an odd point, you're going to get the answer $1$. If you stop it at an even point, you get the answer $0$. Clearly, that's obvious, right? ... So what number are we going to attach to this infinite sum? Do we stop at an odd or an even point? We don't know, so we take the average of the two. So the answer's a half."

Next:

$S_2 \ \ = 1 - 2 + 3 - 4 + \cdots$

$S_2 \ \ = \ \ \ \ \ \ \ 1 - 2 + 3 - 4 + \cdots$

Adding the above two lines, we get:

$2S_2 = 1 - 1 + 1 - 1 + \cdots$

Therefore, $2S_2=S_1=\frac{1}{2}$ and so $S_2=\frac{1}{4}$.

Finally, take

\begin{align}
S - S_2 & = 1 + 2 + 3 + 4 + \cdots
\\ & - (1 - 2 + 3 - 4 + \cdots)
\\ & = 0 + 4 + 0 + 8 + \cdots
\\ & = 4 + 8 + 12 + \cdots
\\ & = 4S
\end{align}

Hence $-S_2=3S$ or $-\frac{1}{4}=3S$.

And so $S=-\frac{1}{12}$. $\blacksquare$