  Our main goal is to show that the Gelfand--Tsetlin toric degeneration of the
type A flag variety can be obtained within a degenerate
representation-theoretic framework similar to the theory of PBW degenerations.
In fact, we provide such frameworks for all Gr\"obner degenerations
intermediate between the flag variety and the GT toric variety. These
degenerations are shown to induce filtrations on the irreducible
representations and the associated graded spaces are acted upon by a certain
associative algebra. To achieve our goal, we construct embeddings of our
Gr\"obner degenerations into the projectivizations of said associated graded
spaces in terms of this action. We also obtain an explicit description of the
maximal cone in the tropical flag variety that parametrizes the Gr\"obner
degenerations we consider. In an addendum we propose an alternative solution to
the problem which relies on filtrations and gradings by non-abelian ordered
semigroups.
