(I know this is an old post, but maybe this'll be of interest to future searchers.)

As far as Fomin--Kirillov's purposes are concerned, the Bruhat representation you mention is all that's really important, and my impression is that they restricted to the quadratic relations for simplicity and generality since the quadratic relations were enough for their purposes.

A little more explicitly, Fomin--Kirillov's main argument is that under the Bruhat representation acting on Schubert polynomials, the Dunkl element $\theta_j$ is just multiplication by $x_j$ via Monk's formula. Hence applying an element of the positive cone $\mathcal{E}_n^+$ under the Bruhat representation to a Schubert polynomial gives a manifestly positive sum of Schubert polynomials. Fomin--Kirillov wanted to get a Schubert-Schubert rule, so we try to take advantage of the fact that

$$ \mathfrak{S}_v(x_1, \ldots, x_n) \mathfrak{S}_w(x_1, \ldots, x_n) = \mathfrak{S}_v(\theta_1, \ldots, \theta_n) \cdot \mathfrak{S}_w(x_1, \ldots, x_n).$$

However, you really need commutativity of the $\theta_j$'s for the right-hand side to be well-defined. (You could use Billey-Jockush-Stanley to expand $\mathfrak{S}_v$ in the standard monomial basis, for instance, but commutativity is clearly desirable, and that's what FK went for.) The real work in Fomin--Kirillov is proving this commutativity statement, and for that they only ended up needing the quadratic relations from the Bruhat representation. For what it's worth Postnikov's quantum pieri rule follow-up and the recent work of Meszaros et al using the Fomin--Kirillov approach all just uses the quadratic relations. That said, Meszaros et al's rules are quite complicated--perhaps they'd be simpler if, say, cubic relations had been allowed?

(Again, this is all just my impression. You could always ask Fomin or Kirillov.)

(I know this is an old post, but maybe this'll be of interest to future searchers.)

As far as Fomin--Kirillov's purposes are concerned, the Bruhat representation you mention is all that's really important, and my impression is that they restricted to the quadratic relations for simplicity and generality since the quadratic relations were enough for their purposes.

A little more explicitly, Fomin--Kirillov's main argument is that under the Bruhat representation acting on Schubert polynomials, the Dunkl element $\theta_j$ is just multiplication by $x_j$ via Monk's formula. Now applying an element of the positive cone $\mathcal{E}_n^+$ under the Bruhat representation to a Schubert polynomial gives a manifestly positive sum of Schubert polynomials. Fomin--Kirillov wanted to get a Schubert-Schubert rule, so we try to take advantage of the Monk's rule fact that

$$ \mathfrak{S}_v(x_1, \ldots, x_n) \mathfrak{S}_w(x_1, \ldots, x_n) = \mathfrak{S}_v(\theta_1, \ldots, \theta_n) \cdot \mathfrak{S}_w(x_1, \ldots, x_n).$$

However, you really need commutativity of the $\theta_j$'s for the right-hand side to be well-defined. (You could use Billey-Jockush-Stanley to expand $\mathfrak{S}_v$ in the standard monomial basis, for instance, but commutativity is clearly desirable, and that's what FK went for.) The real work in Fomin--Kirillov is proving this commutativity statement, and for that they only ended up needing the quadratic relations from the Bruhat representation. For what it's worth Postnikov's quantum Pieri rule follow-up and the recent work of Meszaros et al using the Fomin--Kirillov approach all just uses the quadratic relations. That said, Meszaros et al's rules are quite complicated--perhaps they'd be simpler if, say, cubic relations had been allowed?

One last addendum: Sottile--Bergeron use operators quite similar to the Bruhat operators in "A monoid for the Grassmannian Bruhat order," and they're able to give a monoid with only quadratic and cubic relations for which the operators yield a faithful representation. Maybe something similar holds for the Bruhat representation.

(Again, this is all just my impression. You could always ask Fomin or Kirillov.)