I have been trying to understand manifold theory for a long time now, and every single example of charting a manifold I see is after giving the structure of a connection on it. For instance , consider Visual Differential Geometry chapter-2 in Tristan Needham where he shows the various projections of sphere onto plane, and finally, charting the pseudo sphere.
The issue I have with this is, when we say how the manifold actually sits in space, we have already specified a connection. Actually if one thinks about it, it seems impossible to chart a manifold without specifying a connection through geometrical method as without it, we don't have a concrete entity to draw lines and angles on! (Eg: If we remove the connection coefficients aspect, then sphere and ellipse are same as topological manifolds acc. to the Schuller lectures)
Could I be directed to a source which gives an explicit explanation of how such a construction without any geometric concreteness could be done? It seems impossible to me.
Absolutely not. A manifold is first of all a set of points. On top of that we have a topology. On top of that there is an atlas of charts. (Equivalently, one can start with a set, and a bunch of charts, and then use that to define the topology). A connection has nothing to do with this at all. A connection is about how to move vectors from a vector space attached at one point in the manifold to a vector space attached to a different point of the manifold. By this very heuristic definition, and the order of presentation it should already be clear logically that connections have nothing at all to do with the definition of a manifold or its charts.
I'm not sure I understand your objection to the sphere. $S^n$ is first of all just a set of points. The most common definition is that it is a certain subset of $\Bbb{R}^{n+1}$. Also, regarding the comments of Frederic Schuller regarding the sphere and ellipse being the same, all that means is the two spaces are homeomorphic (and also diffeomorphic once we equip both spaces with their natural differentiable structures). What is true is that it is impossible to talk about lengths and angles on such manifolds without a metric tensor, and it is impossible to talk about "straightness" without specifying a connection.
Anyway, if you don't like spheres, consider the following. Define a relation $\sim$ on $\Bbb{R}^{n+1}\setminus\{0\}$ by saying $x\sim y$ if and only if there exists a $\lambda\in\Bbb{R}$ such that $x=\lambda y$. You can easily verify this is an equivalence relation. Consider the quotient set of all equivalence classes. This is denoted $\Bbb{R} P^n:= (\Bbb{R}^{n+1}\setminus\{0\})/\sim$, and is called the ($n$-dimensional) real projective space.
Note that $\Bbb{R} P^n$ is not defined as a subset of some cartesian space, rather it is a quotient space. On this space, one can define the quotient topology, so now we have a topological space. We can go further and define a smooth structure as follows. For each $i\in\{1,\dots, n+1\}$, let $U_i:=\{[(x^1,\dots, x^{n+1})]\in\Bbb{R} P^n\,:\, x^i\neq 0\}$, and define $\phi_i:U_i\to \Bbb{R}^n$ as \begin{align} \phi_i([x^1,\dots, x^{n+1}])&=\left(\frac{x^1}{x^i}, \dots, \frac{x^{i-1}}{x^i}, \frac{x^{i+1}}{x^i},\dots, \frac{x^{n+1}}{x^i}\right) \end{align} The map $\phi_i$ is well-defined due to the nature of the equivalence relation. The claim is that $\{(U_i,\phi_i)\}_{i=1}^{n+1}$ forms a $C^{\infty}$ atlas for $\Bbb{R} P^n$. Therefore, the maximal atlas containing this is what defines the smooth structure.
Here, $\Bbb{R} P^n$ is most naturally presented as a quotient space, NOT a subset of $\Bbb{R}^k$ for some $k$. Also, I didn't need a connection at all to define the charts (again that would be absurd).