That's wrong.
The "average of the two" (0 and 1 = ½, for S1) is wrong, you cannot do so because the function is not continuous. (The rule they're using only works for continuous functions.)
Secondly, the S2 is not proven, both because it relies on the fraudulent S1 and because the "shifted form" is not proven to be equal to the actual series (1-2+3-4+5-6...) as it could be noted that the negative n is one less than the next n, this would make the series 1+1+1+1+1...
Thirdly, the S that is "solved for" depends on both S1 & S2 which, as we've seen, cannot be concluded are as asserted.
There are more advanced topics in mathematics such as 'analytic continuation of functions', the Riemann-Zeta function, etc. which I can't fully comprehend that suggest that in some sense the sum of all positive integers is -1/12.
The physicist who did the equations claims that since they can use this weird summation result in String Theory that it must be right.
I'm more leaning toward the idea that if that's what they have to rely on to make sense of String Theory, then there's probably something wrong with String Theory.