From the example notations that he gives, all three Einstein notations as well as the Penrose notation make the indices explicit in a way where a mismatch or misalignment will stand out.
Another good example is the Liebnitz notation for derivatives. Proper application of the chain rule visually resembles how fractions cancel: dy/dz dz/dx = dy/dx. It's very easy for the eye to follow and make sure that the cancellation is valid. Newton's notation doesn't make that as easy.
Longhand matrix notation itself is quite good at this. It all but guarantees that you've specified the right number of components in the matrix, and it plays to human strengths in making it easy to check that a matrix is e.g. diagonal or upper-triangular.
I'm going to claim that string diagrams have reasonably good error-detection properties, too, again because they lay facts out in space in a way that humans are quite well optimised for.
Another good example is the Liebnitz notation for derivatives. Proper application of the chain rule visually resembles how fractions cancel: dy/dz dz/dx = dy/dx. It's very easy for the eye to follow and make sure that the cancellation is valid. Newton's notation doesn't make that as easy.