I don't understand why Ramanujan summation is taken seriously

One of the most (in)famous "proofs" that the sum of all natural numbers is -1/12 uses as one of its steps a so-called Ramanujan summation to declare that the the infinite sum

1 - 1 + 1 - 1 + 1 - 1 + ... = 1/2

I don't see anything that would justify this equality. The infinite sum is divergent, ie. it does not converge to any particular value no matter how far you advance in it. There is no justification to assigning the arbitrary value 1/2 to it. (There is no justification to just declare that the result of the entire sum is the average of two consecutive partial results. There's even less justification for it because what you get as partial results depends on how you group the terms.)

How this kind of thing should normally be handled is like this:

1) Hypothesis: The infinite sum 1 - 1 + 1 - 1 + ... = 1/2

2) Counter-proof: We assume that the hypothesis is true and show that it leads to a contradiction. Namely: The assumption leads to the contradictory statement that the sum of all natural numbers, which is divergent, is a particular finite value, which would imply the sum would be convergent. An infinite sum cannot be both divergent and convergent at the same time, thus it's a mathematical contradiction.

3) Thus: The hypothesis cannot be true.

In general, whenever a statement leads to a contradiction, it proves that the statement is false. In this case, we have proven by contradiction the Ramanujan summation as incorrect.

But rather than declaring said summation as incorrect (because it leads to a contradictory result), instead mathematicians have taken it as correct and subsequently also the nonsensical statement that results from it.

It's incomprehensible.