While all other tokens are considered, the attention mechanism is putting an individual weight on each one, in a way "paying more attention" to some than others.

Yup, if you follow this definition of attention. It makes sense.

The mixing step, is computed with the trained weights, meaning the model does learn on its own, when/what to emphasise in this "mixing" process.

Hypothetically speaking, if a token does not matter, it mixing weights could end up being literally 0 (or something close to it)

Most LLM architectures use “soft attention” where some fractional amount of attention is put on every token. “Hard attention” is the term for what you describe.

Softmax takes a vector of arbitrary values and converts it into probabilities such that all the elements of the vector add to 1. Transformers use this to decide on the fractions.

This could be seen as "soft attention" where in theory there would be few winners for every "attention head".

It's also possible that the only purpose the softmax actually serves (or at least major component of) is that of normalization. Without it, the variance in the internal layer dynamics between training samples would be fairly large (some training samples may have tokens that interact heavily, while others not). Making optimization problematic.