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The result is true and in fact we do not need the condition $L(\alpha_0)\leq 1$ since a stronger result is true:

Theorem 1. If $(M,g)$ is a closed simply-connected Riemannian manifold, then there is a constant $C\geq 1$ such that for every closed curve $\alpha_0$ of finite length $L<\infty$, there is a homotopy $\alpha_s$, $0\leq s\leq 1$ between $\alpha_0$ and a constant curve $\alpha_1$ such that $L(\alpha_s)\leq CL$ for all $0\leq s\leq 1$.

Since a reparametrization of a curve does not change its length, we can assume that the curve of length $L$ is parametrized by the constant speed which makes the curve $L/(2\pi)$-Lipschitz as a mapping from $\mathbb{S}^1$ (with arc-length distance) to $(M,g)$. Now the following result is a straightforward consequence of Theorem 5.1 in [1].

Theorem 2. If $M$ is a compact connected and simply-connected Riemannian manifold, then there is $\gamma>0$ such that every $L$-Lipschitz mapping $\alpha:\mathbb{S}^1\to M$ admits a $\gamma L$-Lipschitz extension $A:\overline{\mathbb{B}}^2(0,1)\to M$.

Indeed, using terminology from [1], $M$ is $1$-Lipschitz connected in the small, because it has a finite covering by balls that are bi-Lipschitz homeomorphic to Euclidean balls.

Now, as we pointed out at the beginning, we can assume that $\alpha_0$ is $L/(2\pi)$-Lipschitz. Therefore the extension $A$ is $\gamma L/(2\pi)$ Lipschitz and restrictions of $A$ to circles of radii $0\leq t\leq 1$ will give us a desired homotopy: the restriction to the circle of radius $t$ will have length at most $\gamma Lt\leq \gamma L$. This completes the proof of Theorem 1.

[1] Lang, U., Schlichenmaier, T., Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. Int. Math. Res. Not. 2005, no. 58, 3625-3655. https://arxiv.org/abs/math/0410048

The result is true and in fact we do not need the condition $L(\alpha_0)\leq 1$ since a stronger result is true:

Theorem 1. If $(M,g)$ is a closed simply-connected Riemannian manifold, then there is a constant $C\geq 1$ such that for every closed curve $\alpha_0$ of finite length $L<\infty$, there is a homotopy $\alpha_s$, $0\leq s\leq 1$ between $\alpha_0$ and a constant curve $\alpha_1$ such that $L(\alpha_s)\leq CL$ for all $0\leq s\leq 1$.

Since a reparametrization of a curve does not change its length, we can assume that the curve of length $L$ is parametrized by the constant speed which makes the curve $L/(2\pi)$-Lipschitz as a mapping from $\mathbb{S}^1$ (with arc-length distance) to $(M,g)$. Now the following result is a straightforward consequence of Theorem 5.1 in [1].

Theorem 2. If $M$ is a compact connected and simply-connected Riemannian manifold, then there is $\gamma>0$ such that every $L$-Lipschitz mapping $\alpha:\mathbb{S}^1\to M$ admits a $\gamma L$-Lipschitz extension $A:\overline{\mathbb{B}}^2(0,1)\to M$.

Indeed, using terminology from [1], $M$ is $1$-Lipschitz connected in the small, because it has a finite covering by balls that are bi-Lipschitz homeomorphic to Euclidean balls.

Now, as we pointed out at the beginning, we can assume that $\alpha_0$ is $L/(2\pi)$-Lipschitz. Therefore the extension $A$ is $\gamma L/(2\pi)$ Lipschitz and restrictions of $A$ to circles of radii $0\leq t\leq 1$ will give us a desired homotopy: the restriction to the circle of radius $t$ will have length at most $\gamma Lt\leq \gamma L$. This completes the proof of Theorem 1.

[1] Lang, U., Schlichenmaier, T., Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. Int. Math. Res. Not. 2005, no. 58, 3625-3655.

Since a reparametrization of a curve does not change its length, we can assume that the curve of length $L$ is parametrized by the constant speed which makes the curve $L/(2\pi)$-Lipschitz as a mapping from $\mathbb{S}^1$ (with arc-length distance) to $(M,g)$. Now the following result is a straightforward consequence of Theorem 5.1 in [1].

Theorem. If $M$ is a compact connected and simply-connected Riemannian manifold, then there is $\gamma>0$ such that every $L$-Lipschitz mapping $\alpha:\mathbb{S}^1\to M$ admits a $\gamma L$-Lipschitz extension $A:\overline{\mathbb{B}}^2(0,1)\to M$.

Indeed, using terminology from [1], $M$ is $1$-Lipschitz connected in the small, because it has a finite covering by balls that are bi-Lipschitz homeomorphic to Euclidean balls.

Now, as we pointed out at the beginning, we can assume that $\alpha_0$ is $L/(2\pi)$-Lipschitz. Therefore the extension $A$ is $\gamma L/(2\pi)$ Lipschitz and restrictions of $A$ to circles of radii $0\leq t\leq 1$ will give us a desired homotopy: the restriction to the circle of radius $t$ will have length at most $\gamma Lt\leq \gamma L$.

[1] Lang, U., Schlichenmaier, T., Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. Int. Math. Res. Not. 2005, no. 58, 3625-3655. https://arxiv.org/abs/math/0410048

The result is true and in fact we do not need the condition $L(\alpha_0)\leq 1$ since a stronger result is true:

Theorem 1. If $(M,g)$ is a closed simply-connected Riemannian manifold, then there is a constant $C\geq 1$ such that for every closed curve $\alpha_0$ of finite length $L<\infty$, there is a homotopy $\alpha_s$, $0\leq s\leq 1$ between $\alpha_0$ and a constant curve $\alpha_1$ such that $L(\alpha_s)\leq CL$ for all $0\leq s\leq 1$.

Since a reparametrization of a curve does not change its length, we can assume that the curve of length $L$ is parametrized by the constant speed which makes the curve $L/(2\pi)$-Lipschitz as a mapping from $\mathbb{S}^1$ (with arc-length distance) to $(M,g)$. Now the following result is a straightforward consequence of Theorem 5.1 in [1].

Theorem 2. If $M$ is a compact connected and simply-connected Riemannian manifold, then there is $\gamma>0$ such that every $L$-Lipschitz mapping $\alpha:\mathbb{S}^1\to M$ admits a $\gamma L$-Lipschitz extension $A:\overline{\mathbb{B}}^2(0,1)\to M$.

Indeed, using terminology from [1], $M$ is $1$-Lipschitz connected in the small, because it has a finite covering by balls that are bi-Lipschitz homeomorphic to Euclidean balls.

Now, as we pointed out at the beginning, we can assume that $\alpha_0$ is $L/(2\pi)$-Lipschitz. Therefore the extension $A$ is $\gamma L/(2\pi)$ Lipschitz and restrictions of $A$ to circles of radii $0\leq t\leq 1$ will give us a desired homotopy: the restriction to the circle of radius $t$ will have length at most $\gamma Lt\leq \gamma L$. This completes the proof of Theorem 1.

[1] Lang, U., Schlichenmaier, T., Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. Int. Math. Res. Not. 2005, no. 58, 3625-3655. https://arxiv.org/abs/math/0410048

Since a reparamteization of a curve does not change its length, we can assume that the curve of length $L$ is parametrized by the constant speed which makes the curve $L/(2\pi)$-Lipschitz as a mapping from $\mathbb{S}^1$ (with arc-length distance) to $(M,g)$. Now the following result is a straightforward consequence of Theorem 5.1 in [1].

Theorem. If $M$ is a compact connected and simply-connected Riemannian manifold, then there is $\gamma>0$ such that every $L$-Lipschitz mapping $\alpha:\mathbb{S}^1\to M$ admits a $\gamma L$-Lipschitz extension $A:\overline{\mathbb{B}}^2(0,1)\to M$.

Indeed, using terminology from [1], $M$ is $1$-Lipschitz connected in the small, because it has a finite covering by balls that are bi-Lipschitz homeomorphic to Euclidean balls.

Now, as we pointed out at the beginning, we can assume that $\alpha_0$ is $L/(2\pi)$-Lipschitz. Therefore the extension $A$ is $\gamma L/(2\pi)$ Lipschitz and restrictions of $A$ to circles of radii $0\leq t\leq 1$ will give us a desired homotopy: the restriction to the circle of radius $t$ will have length at most $\gamma Lt\leq \gamma L$.

[1] Lang, U., Schlichenmaier, T., Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. Int. Math. Res. Not. 2005, no. 58, 3625-3655. https://arxiv.org/abs/math/0410048

Since a reparametrization of a curve does not change its length, we can assume that the curve of length $L$ is parametrized by the constant speed which makes the curve $L/(2\pi)$-Lipschitz as a mapping from $\mathbb{S}^1$ (with arc-length distance) to $(M,g)$. Now the following result is a straightforward consequence of Theorem 5.1 in [1].

Theorem. If $M$ is a compact connected and simply-connected Riemannian manifold, then there is $\gamma>0$ such that every $L$-Lipschitz mapping $\alpha:\mathbb{S}^1\to M$ admits a $\gamma L$-Lipschitz extension $A:\overline{\mathbb{B}}^2(0,1)\to M$.

Indeed, using terminology from [1], $M$ is $1$-Lipschitz connected in the small, because it has a finite covering by balls that are bi-Lipschitz homeomorphic to Euclidean balls.

Now, as we pointed out at the beginning, we can assume that $\alpha_0$ is $L/(2\pi)$-Lipschitz. Therefore the extension $A$ is $\gamma L/(2\pi)$ Lipschitz and restrictions of $A$ to circles of radii $0\leq t\leq 1$ will give us a desired homotopy: the restriction to the circle of radius $t$ will have length at most $\gamma Lt\leq \gamma L$.

[1] Lang, U., Schlichenmaier, T., Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. Int. Math. Res. Not. 2005, no. 58, 3625-3655. https://arxiv.org/abs/math/0410048