There are two main points of confusion in your reasoning.

1. Any field can be decomposed as a sum of two fields: one with zero divergence and one with zero rotor.
   You say any field can be constructed using its divergence and rotational which is kind of obscure to me.

2. If a field has zero divergence and zero rotor, it can still be finite! It does not have to be zero, if its divergence and rotor are zero.

So to sum up, the magnetic field outside of the wire has zero divergence (as everywhere) and zero rotor, but it is finite.

There are two main points of confusion in your reasoning:

1. Any field can be decomposed as a sum of two fields: one with zero divergence and one with zero rotor.
   You say "any field can be constructed using its divergence and rotational" which is kind of obscure to me.

2. If a field has zero divergence and zero rotor, it can still be finite! It does not have to be zero, if its divergence and rotor are zero.

So to sum up, the magnetic field outside of the wire has zero divergence (as everywhere) and zero rotor, but it is finite.