Smoothness of components of the Emerton-Gee stack for ${\text{GL}}_2$

Let $K$ be a finite unramified extension of ${\mathbb{Q}}_p$, where $p>2$. [CEGS22b] and [EG23] construct a moduli stack of two dimensional mod $p$ representations of the absolute Galois group of $K$. We show that most irreducible components of this stack (including several non-generic components) are isomorphic to quotients of smooth affine schemes. We also use this quotient presentation to compute global sections on these components.