I want a property of a Riemannian manifold $(M,g)$ that excludes the possibility that a closed curve of arbitrarily short length is not homotopic to a point. What is this property?
An example of a manifold which does not satisfy my requirement is the revolution of $(x,1/x)$ for $x>0$ around the $x$-axis in $\mathbb{R}^3$. I am not sure if the deforming the closed curve to a point in the limit $x\rightarrow\infty$ counts? If so, I guess it is possible to come up with some other example.
Attempt: I think it suffices that the injectivity radius $r(x):M\rightarrow[0,\infty)$ satisfies $\inf_{x\in M}r(x)>0$. If there exists a sequence of closed geodesics $\gamma_k$ of length $l(\gamma_k)\leq1/k$, then we can pick $x_k,y_k\in\gamma_k$ at a maximal geodesic distance from each other. Then are two different geodesic segments of equal length connecting $x_k$ and $y_k$. For $1/k<R$ this is a contradiction.
However, I don't know how to go from the existence of a closed curve of arbitrarily short length to the existence of a closed geodesic of arbitrarily short length.
Here some elaborations on the comments:
For every point $x$ in the manifold there is a maximal radius $r_x$ so that the ball $B_{r_x}(x)$ is diffeomorphic to $\Bbb R^n$, which is contractible. As such if a curve has length $<r_x$ and hits the point $x$ it will be contained entirely in a contractible neighbourhood and thus also be contractible. One possible criterium for when arbitrarily small curves must be contractible is then that the numbers $\{r_x\mid x\in M\}$ are bounded from below.
An elementary (and topological) criterium guaranteeing this is that the manifold is compact. Note that both completeness of the metric (your example) or totally boundedness (bounded a disk minus a point) alone are not enough.
The radius $r_x$ is larger than the injectivity radius of the point $x$, so another (geometric) criterium is that the injectivity radius is bounded from below. Note that this is impossible if the metric is not complete.
Contractivity of the neighbourhoods is a very strong property, for example if the $r_x$ admit a lower bound you will find that any map $f: X \to M$ is homotopic to a constant map if $f(X)$ has diameter smaller than the lower bound. If you are only interested in loops then maybe this is too much, but this "looking for lower bounds" business might also seem fruitful to you. In this case let $$R(x) := \sup \{ R\in \Bbb R \mid B_R(x) \text{ is simply connected}\},$$ then you may ask $\inf_{x\in M}R(x)>0$, this is equivalent to any curve that is "small enough" being null-homotopic in a small neighbourhood. I'm not sure of any nice ways to gurantee this that are weaker than what I already wrote.