If you're willing to quotient by a nonreductive group, then $M_n//B$ will get you the $GL(n)$ flag manifold.

But that one is also available reductively. Let $V_0,V_1\ldots,V_n$ be a list of vector spaces with those dimensions, and let $Hom := \prod_{i=1}^n Hom(V_{i-1},V_i)$. If we quotient this by $GL(V_1)\times \cdots \times GL(V_{n-1})$, it forgets the actual maps and only remembers the images inside $V_n$, so the result is (or to be precise, can be chosen to be) the manifold of flags in $V_n$. I forget whom this is due to, but it's pretty old.

You can get some of this to work for symplectic and orthogonal groups, using the $O(V) \times Sp(W)$ action on $V\otimes W$; the reference I know is [Lerman-Montgomery-Sjamaar].

If you're willing to quotient by a nonreductive group, then $M_n//B$ will get you the $GL(n)$ flag manifold. (People are usually afraid to do so, worrying that the ring of invariants won't be Noetherian, but this one is.)

That flag manifold is also available reductively. Let $V_0,V_1\ldots,V_n$ be a list of vector spaces with those dimensions, and let $Hom := \prod_{i=1}^n Hom(V_{i-1},V_i)$. If we quotient this by $GL(V_1)\times \cdots \times GL(V_{n-1})$, it forgets the actual maps and only remembers the images inside $V_n$, so the result is (or to be precise, can be chosen to be) the manifold of flags in $V_n$. I forget whom this is due to, but it's pretty old.

You can get some of this to work for symplectic and orthogonal groups, using the $O(V) \times Sp(W)$ action on $V\otimes W$; the reference I know is [Lerman-Montgomery-Sjamaar].