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Just don't reinvent the wheel.

At the end of the day, the common property of any definition of spinor is that it picks up a minus sign under a $2\pi$ rotation. This is a statement about how spinors transform. Representation theory is precisely the language of "how things transform". You can try to use a different set of words in your definition of spinor, of course. But we already have a theory that describes how things transform, representation theory, so every definition of spinor will, at least implicitly, use rep-theoretic notions, even if you decide to use a different terminology.

OP is asking why we need representation theory to understand spinor bundles but not for vector/tensor bundles.

This is similar to asking why we need the full machinery of Riemannian integration to understand the area below the function $f(x)=x^2$, but not for the function $f(x)=x$. The short answer is: linear functions are so simple that you can calculate their area without reference to Riemann partitions etc. But generically, areas do require these notions. For simple functions you can "get away" with this, only because they are so simple you can phrase everything using very primitive notions. But if you want to generalize and unify areas to arbitrary functions, you need to use the correct language.

The formal definition of a spinor is an element of a special type of principal bundle. And general associated vector bundles are defined using the language of representation theory. A tensor bundle is just another special type of vector bundle, just one so simple that you can "get away" with it and define it without (explicit) use of representation-theoretic language. But spinor, and more general, bundles, do require the full machinery of representation theory. Which brings me to the actual answer to the question in the OP:

But you can rephrase everything in a more geometric language, if you will.

If you take a generic manifold, you can define the tangent bundle and various Cartesian powers thereof. You can even restrict to smaller bundles in a consistent way by imposing symmetry/anti-symmetry constraints. For example, a rank-2 tensor can be decomposed into its symmetric and anti-symmetric parts in a way consistent with changes of coordinates.

In rep theory language, a manifold has a natural $GL(n)$ structure and therefore there are natural bundles which transform as representations of $GL(n)$. The symmetry/anti-symmetry of tensors is just the statement that the irreducible representations of $GL(n)$ can be obtained by applying suitable symmetrizers to the fundamental representation, this is the old Schur correspondence between irreps of $S_n$ and $GL_n$. You don't need to phrase this in rep theoretic language, but why would you not? This language makes it clear that there are no further constraints you could impose in a way consistent with generic changes of coordinates, since these representations are all the irreps of $GL(n)$.

If you have a non-degenerate metric, the $GL(n)$ structure group can be restricted to an $O(n)$ group. You can now further decompose your tensors into its trace, symmetric-traceless, and anti-symmetric parts, and this is still compatible with metric-preserving changes of coordinates.

You can again rephrase this in rep theoretic language: all irreps of $O(n)$ can be recovered from irreps of $GL(n)$ by imposing suitable tracelessness constraints.

A key point of $O(n)$ is that, unlike $GL(n)$, this group is not simply-connected, so there are more options for the various bundles you can consider. In particular, $O(n)$ has a (pair of, actually) canonical universal-cover, $Pin(n)$. A spinor field is an element of the corresponding $Pin(n)$ bundle, i.e., it is a section of a double cover of your structure group.

This is the definition of spinor in geometric language. You can phrase this in rep theoretic language by saying that $Pin(n)$ has representations that do not give representations of $O(n)$, they are projective representations instead. This is the exact same claim as above, just with different words. You can define spinors without explicitly using representation theory language, but the theory is 100% there, you are just using different words.