Short answer: No analogue of Conway’s “Mathieu groupoid” for M24 (on 31 holes) exists in the same canonical sense as M13. This was looked at quite seriously (by Conway and others), and there are clean obstructions.

What does work, and why one might hope for it

• The “right” ambient set is indeed V = F2^5 \ {0}, with |V| = 31.
• A “hole” should be a 3-dimensional subspace U ≤ V (a plane in PG(4,2)), which contributes 7 excluded points, leaving 24 points to permute.
• As you observed, this matches a lot of the usual M24-combinatorics:
  • The stabilizer of a 3-space in GL(5,2) is 2^6:(GL(3,2) × GL(2,2)), and in M24 this is a stabilizer of a trio (a partition of the 24 points into three disjoint octads).
  • Fixing U, the 24 points naturally split into three 8-sets (octads) given by the three non-zero cosets modulo U in a chosen 2-dimensional complement. So the dictionary “plane ↔ trio” is exactly right.

Where the attempt fails

• To have a Mathieu groupoid in Conway’s sense you need a canonical “move” φU→W between complements of two holes U and W, depending only on the incidence (here, dim(U ∩ W) = 1 or 2), and equivariant under GL(5,2). Composing such moves along paths with the hole returning to U should generate the desired isotropy M24 on the 24-point complement.
• In contrast to the M13/PGL(3,3) situation, for planes in PG(4,2) there is no GL(5,2)-equivariant way to choose such a bijection φU→W. Equivalently: fix an ordered pair (U,W) with prescribed intersection dimension r ∈ {1,2}. Let H = StabGL(5,2)(U,W). Any canonical rule would give an H-fixed bijection XU → XW between the two 24-point complements. But one checks that H has no fixed point on this set of bijections:
  • For r = 2, H acts nontrivially on the induced partition of XU (and XW) into 3 octads; there is no H-invariant way to identify these partitions.
  • For r = 1, the situation is even worse: the stabilizer acts with more freedom on the two sides, and again there is no H-invariant bijection.
• In short, the pair (U,W) does not determine any canonical bijection between the two 24-point complements, so there is no way to set up “moves” that are both well-defined and GL(5,2)-equivariant. This is the key obstruction, and it kills an M31-analogue in Conway’s sense.

What you can do instead

• The close ties you noticed are real and are the basis of the standard constructions of M24 (e.g., via the Miracle Octad Generator and the hexacode). A fixed plane U does give a trio; M24 acts transitively on trios and their refinements. But this gives a very effective way to build M24, not a groupoid where “moves” are uniquely determined by the endpoints as in M13.

References and pointers

• J. H. Conway and A. J. E. Ryba, The Mathieu groupoid M13, Experimental Mathematics 13 (2004), 223–236. Besides describing M13, the paper and related discussions make clear why the M24 analogue does not materialize.
• R. T. Curtis, A new combinatorial approach to M24, Math. Proc. Cambridge Philos. Soc. 79 (1976), 25–42. The “MOG” (Miracle Octad Generator) and the trio/octad machinery you mentioned.
• J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (3rd ed.), especially the sections on the Golay code, octads, trios, and the GL(5,2) dictionary.
• A. A. Ivanov, The Mathieu Groups, Cambridge Univ. Press, 1998, for systematic treatment of trios, octads, and related stabilizers in M24.

Summary

• The natural 31-hole setup matches stabilizer types and the trio partition, but there is no GL(5,2)-equivariant way to define the “moves” between holes solely from the incidence (dim 1 vs dim 2 intersection). This prevents the existence of a Mathieu groupoid M31 analogous to M13.