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:

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

1. 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 $$e^{\;\;a}_m$$.

2. 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?

• Related question here. Commented Jul 18, 2020 at 19:57
• Minor complaint: it is better to use different alphabets (e.g. Greek vs. Latin or uppercase vs. lowercase) rather than different parts of the same alphabet to distinguish between different types of indices, it is less work for the eyes to sort out indices in complicated expressions. Commented Jul 19, 2020 at 6:10

The YM action for the Poincare group as you write down is perfectly allowable in the effective field theory framework, as long as you double check that pathological tachyons are absent. There are tons of papers devoted to the so called $$f(R)$$ and $$f(T)$$ theories with higher-order Lagrangian terms (like $$R^2$$, $$T^2$$).
The catch is that, comparing with the EH term, the YM term is suppressed by a factor of $$O(p^2/M_p^2)$$, where $$M_p$$ is the Planck mass. Therefore, the YM term is negligible, except in extreme situations, e.g. shortly after the Big Bang.