According to https://kazemnejad.com/blog/transformer_architecture_positio... they explain it as a clever way to satisfy the following criteria:

    - It should output a unique encoding for each time-step (word’s position in a sentence)
    - Distance between any two time-steps should be consistent across sentences with different lengths.
    - Our model should generalize to longer sentences without any efforts. Its values should be bounded.
    - It must be deterministic.

Your example contradicts the 'values should be bounded' criterion as it generalizes to longer sentences.

I'm no expert, but I think it's so that the model can learn the relative position wrt other tokens.

They use indices for models like vision transformers with a fixed number of patches but for variable length context I think it's more beneficial to use encodings that can also capture the relative distance.