Confusion about the primary decomposition theorem in linear algebra

Question

I am currently studying the Jordan canonical form which uses the primary decomposition.

I have seen the generalised eigenspace decomposition and I know that the algebraic multiplicity which appears in the characteristic polynomial is equal to the dimension of the generalised eigenspace. I thought this decomposition is related to the primary decomposition.

So my question is: why do we use the minimal polynomial instead of the characteristic polynomial in the primary decomposition? Or is there a way to prove the Jordan canonical form theorem without using primary decomposition?

Thanks a lot!