I think that your theorem is "almost" true if we restrict ourselves to a neighborhood of x in (some quotient of) the k-skeleton. And indeed, this is a way to give a precise meaning to "holes" in singular homology. Let me be more specific.
Let's consider a finite CW-complex. The space is built starting with a finite number of points, and attaching a finite number of cells of various dimensions. The k-skeleton $X_k$ is the union of all cells of dimension less than or equal to k.
The smash $X_k|X_{k-1}$ is obtained from $X_k$ by identifying all points in $X_{k-1}$. If X is a CW-complex, that "smash" is homeomorphic to a bouquet of k-spheres. These spheres are the "holes" we are looking after. From the standard identification
$H_k(X_k|X_{k-1})simeq H_k(X_k,X_{k-1})$ , we see that each sphere gives rise to an element in the relative homology group
$H_k(X_k,X_{k-1})$. There are two ways these elements may fail to give an element in $H_k(X)$.
--> Instead of capturing a k-dimensional "hole", the cell may in fact "fill" a (k-1)-dimensional "hole". That's what happens when, for example, we cap a cylinder with a disk. So, we are only interested in elements in the kernel of the boundary operator $delta_k : H_{k}(X_k,X_{k-1})rightarrow H_{k-1}(X_{k-1})$.
--> The k-dimensional "hole" may be filled by some $k+1$-dimensional cell. So we should quotient $H(X_k,X_{k-1})$ by the image of the operator $H_{k+1}(X_{k+1},X_k)rightarrow H_{k}(X_k)rightarrow H_{k}(X_k,X_{k-1})$. Let us denote that image by $E_k$.
And it works. The group $H_k(X)$ is actually isomorphic to the quotient $ker delta_k / E_k$. So there is a subset of the "holes" in $X_k|X_{k-1}$ that provide a generating family for $H_k(X_k)$. Arguably, the spherical neighborhood you are looking for exists in the smash, not in X, but still, I think it succeeds in making our intuition rigorous. As a reference, I may point to Greenberg "Algebraic topology, an introductory course" (21.8 ff).
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