This is a good concern. FWIW, though, there ARE natural summability methods which turn mere indexed sequences of terms into sums, without any particular choice

This is a good concern. FWIW, though, there ARE natural summability methods which turn mere indexed sequences of terms into sums, without any particular choice as to how to interpret those sequences as specializations of formal series, which happen to assign the sequence whose n-th term is n a sum of -1/12. For an example of one such method, see the end of my post elsewhere in this thread.