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Divergence does not imply that a sum cannot be analyzed, manipulated, or assigned a value. For example, the sum of Grandi's Series can be shown in numerous rigorous ways, such as Borel Summation (I personally prefer the integration method), Ramanujan summation, Euler regularization (which is roughly equivalent to Abel summation, but the latter is more rigorous because it addresses specific criticisms of Euler's method). The answer 1/2 is unique and consistent. Every method capable of summing the series will obtain the answer 1/2, and no method can obtain a different answer. There are no contradictions, only wrong answers obtained by violating rules (the common argument of grouping terms to produce 0 or 1 violates the Rearrangement Theorem and is one of the earliest known examples of the Eilenburg-Mazur Swindle).
Manipulations of divergent sums are not illegal, they are simply not strongly justified by the normal axioms of finite arithmetic. The additive identity of zero, for example, is not illegal, it just requires stronger justification. This is why it arrives at the unique, consistent answer for some divergent sums but produces contradictions in others. As a general rule, zeroes can be added in divergent sums that can be written as an equivalent Dirichlet series. Specifically, using the Dirichlet transform is the main method of zeta function regularization.
Most importantly, you cannot apply the intuition of finite arithmetic to infinite sums: even convergent ones. "You can't add positive numbers and get a negative answer" is an example of this reasoning, as is "You can't add whole numbers and get a fraction". These finite axioms can be easily shown to be irrelevant in the world of infinite sums:
1) First, a simple proof that a finite sum of rational numbers can never equal an irrational number: For any rational numbers a1, a2, a3,...an, then a1+a2+a3+...+an=(a1+a2+a3+...+an)/(a1*a2*a3*...*an). Since this number is expressible as a fraction of whole numbers, it is rational.
2) Second, a very important and powerful example that you yourself used in this article: ζ(2)=π²/6. This is an irrational number, but every term of the series ζ(2) is a rational number. If it were a finite sum, it would be impossible to obtain an irrational answer. But this sum isn't even divergent, it is convergent and well-behaved. So the breakdown of our finite rules is not because of divergence, but because we are not taking finite sums as we are used to doing.
The value of 1+2+3+4+.... is -1/12, and just because Numberphile did not explain the more complex justifications for their manipulations does not make those manipulations invalid. The result can be proven in many more difficult and abstract ways if that appeals to you more; I would recommend starting with Terence Tao's article on the subject.