When we define addition by cardinality we require a lot of machinery to be already in place.
We want to add 2 + 3 = 5.
We need to have 2,3, 5 as numbers.
We need initial counting segments or counting head segments from 1 to 2, 1 to 3 and 1 to 5.
We need to be able to find sets without common elements in bijection with the counting head segments 2 and 3.
We need to be able to do a bijection with the counting head segment 5.
When we do the addition identities, we have to do even more.