By definition, a strictly semistable model $\mathscr{X}$ of $X$ is proper over ${K^{\circ}}$. Using that $X$ is a curve, we will deduce that $\mathscr{X}$ is projective over ${K^{\circ}}$. Indeed, the special fiber $\mathscr{X}_{s}$ is a proper curve over the residue field and hence projective. It is easy to construct an effective Cartier divisor $D$ on $\mathscr{X}$ whose support intersects any irreducible component of $\mathscr{X}_{s}$ in a single closed point. By [Liu06, Exercise 7.5.3], the restriction of $D$ to $\mathscr{X}_{s}$ is ample. It follows from [EGAIV, Cor. 9.6.4] that $D$ is ample and hence $\mathscr{X}$ is projective.

Similarly, we can define strictly semistable formal models of ${X^{{\mathrm{an}}}}$. Using that $X$ is a smooth projective curve, the algebraization theorem of Grothendieck [EGAIII, Thm. 5.4.5] and its generalizations to the non-noetherian setting [Abb11, Cor. 2.13.9], [FK88, Prop. I.10.3.2] show that formal completion induces an equivalence of categories between strictly semistable algebraic models of $X$ and strictly semistable formal models of ${X^{{\mathrm{an}}}}$. Here, we need a similar argument as above to construct an effective formal Cartier divisor which restricts to an ample Cartier divisor on the special fiber.