Why Einstein action is not Yang-Mills action for gauge theory of Poincaré algebra?

It is well known, how to construct Einstein gravity as gauge theory of Poincare algebra. See for example General relativity as a gauge theory of the Poincaré algebra.

There are

1. Construction of covariant derivative:

$$ nabla_m = partial_m -i e_m^{;a}P_a -frac{i}{2}omega_m^{;;;cd}M_{cd}.$$

2. Impose covariant constraint on geometry:
   $$
   [nabla_m, nabla_n] = -i R_{mn}^{;;;a}P_a -frac{i}{2}R_{mn}^{;;;ab}M_{ab}
   $$
   $$
   R_{mn}^{;;;a} = 0.
   $$
   From this equation, spin connection $ω^{;;;cd}_m$ is expressed in terms of veilbein $e^{;;a}_m$.

3. Now, one can easily construct Einstein-Hilbert action:
   $$
   S_{EH} = int d^d x e ;R_{mn}^{;;;ab} e_a^{;m}e_b^{;n}
   $$
   $e_a^{;m}$ is inverse veilbein $e_a^{;m} e_m^{;b}= delta_a^b $. Metric tensor:
   $$
   g_{mn} = e_m^{;a}e_n^{;b} eta_{ab}.
   $$

But one can modify second step and obtain another actions, with additional dynamical spin connection:

1. $$
   S_{EH} = int d^d x e ;R_{mn}^{;;;ab} e_a^{;m}e_b^{;n}.
   $$

2. $$
   S_{YM} = int d^d x e left(;R_{mn}^{;;;ab} R_{kl}^{;;;cd}g^{mk}g^{nl}eta_{ad}eta_{bc} + R_{mn}^{;;;a} R_{kl}^{;;;b}g^{mk}g^{nl}eta_{ab}right).
   $$

So I have few questions:

What will standard Einstein-Hilbert action describe in this case?

What is Yang-Mills theory for Poincare group? Which properties have such theory?

Why Einstein action is not Yang-Mills theory for Poincare group?