Pretty much every time I see someone write down an action and find it's stationary points they immediately switch to coordinates and expand the Euler-Lagrange equations. Specifically, take the Einstein Hilbert action:
$$S=\int_M\langle Rc,g\rangle _g \text{dvol}_g$$ One usually writes down the volume form in coordinates, and the Ricci curvature in coordinates, then does some manipulation with Christoffel symbols to obtain the field equations: $$Rc_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=0$$ but why can we do this? I get that these objects are coordinate independent, but the integral is over all of $M$, so how does varying the action in a chart lead us to conclude that this is a global equation for $g$? Unless for some reason the integral should vanish on an arbitrary compact subset of a coordinate patch of $M$ I don't really see how to justify this, and I haven't seen a text which justifies this, but maybe it is trivial and I am just being dumb. Any help would be appreciated.
In general, this can have many reasons and I think this question is not dumb at all. Most common ones are:
Your manifold can be covered by $1$ chart, this for example includes space-time $\mathbb{R}^{3+1}$. In this case, you can choose global coordinates, in particular a nice volume form, and using your definition of integration on manifolds, the integrals coincide.
The next possibility is that there exists a globally defined object, which gives rise to something coordinate independent as you noted. In your case, everything is coordinate free: Curvature, metric tensor and a volume form. You just need to take care of the overlaps between two charts somehow. Hence by abuse of notation, you just write it in local coordinates under the integral. Intrinsic description are sometimes rather clunky and advanced, but they exist.
An example: Would you call the term $|d f |^2$ for a map $f:M \to N$ between manifolds a smooth section of the bundle $T^*M \otimes f^*TN \to M$ with metric induced by the metric on $g^{-1}$ on $T^* M$ and $h$ on $TN$? Or just write down some local coordiante expression involving $2$ metric tensors with index lowering/raising.
The last reason is: You really want to look at something locally. Some arguments, like regularity theory, only require you to look at solutions of certain PDEs locally. In this case, it is also often not mentioned and is often obscured, since global results are much harder to come by (and often understand). As also pointed out in the comments, the Euler-Lagrange equation are a point wise identity, so a local description (in some open subset) is enough.