I think that the above action exists (of course presupposing a sufficiently nice base-space $X$) and is given by $$Y\curvearrowleft\pi_1(X,x)\rightarrow Y, [\Gamma].[\gamma]=[\Gamma\circ\gamma]$$ where $x$ is a fixed point in the base space $X$, and $Y$ is the universal covering space modeled as the set of path-classes in $X$ which start at $x$. In the above, $\Gamma$ is any such path and $\gamma$ is an element of the fundamental group, i.e. w.l.o.g. a loop starting at $x$ and what I wrote is well-defined.
I think it is immediate that the above has the defining properties of a group action, but I can't find this result when googling so I'm not sure if maybe I did something wrong. Can someone confirm that this is indeed an action?
Bonus question: Why is this an action by isometries for compact, connected Riemannian Manifolds? I know how the metric on the universal cover is created but am not able to do the necessary calculations yet.
Your definition of this action does not make sense because it violates the definition of path concatenation. Here's why.
$\Gamma$ represents any path that starts at $x$. Let's consider the case that $\Gamma$ ends at a point $z \ne x$.
$\gamma$ represents a path that starts and ends at $x$.
In order for the concatenation $\Gamma \circ \gamma$ to be defined, the endpoint of $\Gamma$ (which is $z$) must equal the starting point of $\gamma$ (which is $x$). Since $z \ne x$, it follows that the concatenation $\Gamma \circ \gamma$ is not defined.