Nothing “nice’’ has ever turned up in the literature beyond the naïve linear‐polynomial cut‐out that you describe. In fact if one identifies F = 𝔽_{2^{11}} ≅ 𝔽₂[x]/(m(x)) and writes an additive endomorphism of F as

f(x) = a₀x + a₁x² + a₂x^{2²} + … + a_{10}x^{2^{10}},

then the condition “f permutes the cyclic subgroup C of order 23” is equivalent to

(1) a_i^{2^{11}} − a_i = 0, i=0,…,10, (forcing a_i∈F),
(2) ∏_{c∈C} ( t − f(c) ) = t^{23} − 1 in F[t].

When you expand the second line you get exactly 23 polynomial equations in the 11 unknowns a₀,…,a_{10}, each of degree ≤23. Together with the eleven Frobenius‐fixing equations you have 34 equations cutting out M₂₃ as a zero–dimensional algebraic subgroup of GL(11,2).

A priori one hopes that a clever choice of generators or a Gröbner‐basis run in Magma or Singular will simplify these 34 equations, but in practice they remain unwieldy. No substantially more compact “magic’’ system of polynomial equations for M₂₃ is known.

In the literature one almost always passes instead to one of the classical realizations of M₂₃:

• The stabilizer of one coordinate in M₂₄, the automorphism group of the binary Golay code of length 24;
• The automorphism group of the Steiner system S(4,7,23), embedded as a subgroup of S₂₃;
• The permutation representation on 23 points together with the usual Coxeter‐type presentation by three generators of orders 2, 3, 23 and a small list of braid‐relations.

Each of those approaches gives you either a matrix–group embedding (and hence a huge collection of quadratic equations coming from “gᵀBg = B’’ for some form B, or from “g_i g_j … g_k = e’’) or group‐theoretic relations, but not a dramatically shorter system of polynomial equations in the 11 parameters a_i.

So, to answer your question:
– Yes, the additive‐polynomial construction you wrote down is exactly the natural “algebraic‐set’’ description of M₂₃ in GL(11,2),
– No, nobody has produced a substantially simpler set of defining polynomials, nor is there a published Gröbner‐basis that collapses those 34 equations to something of any reasonable size.

If you want a more “compact’’ algebraic description you will have to choose one of the well–known embeddings (Golay‐code or Steiner‐system or Coxeter‐presentation) and live with the resulting quadratic or higher–degree relations in the matrix–entries or generators. But a truly small system of polynomials in your 11 additive‐polynomial coefficients simply is not known.