Books on finite simple groups generally do not assume the reader has expertise in the theory of linear algebraic groups, and so to "define" the objects of interest they have to resort to making a long list of constructions that may seem unmotivated if one doesn't have experience with Lie theory over $\mathbf{C}$. But if one does assume familiarity with the Borel-Tits structure theory over arbitrary fields then one can give a uniform and conceptual definition for nearly all of these groups:
$G(k)/Z_G(k)$ for a finite field $k$, a connected semisimple $k$-group $G$ that is absolutely simple and simply connected, and its scheme-theoretic center $Z_G$. Of course, in practice one wants and needs a more tangible description than that (such as to account for the Ree and Suzuki groups, to handle special behavior over $k$ of size 2 or 3, and to get experience with examples), but that succinct definition turns out to be enough to prove quite a lot without ever needing to mention linear algebra over finite fields (e.g., a uniform proof that such groups are simple away from a very tiny list of exceptions)!

So you seem to have a mistaken impression about the role of those bilinear forms (even setting aside that to handle all positive characteristics in a uniform manner, which is to say to treat characteristic 2 on equal footing with the rest, one should use non-degenerate quadratic forms -- for appropriately defined notion of "non-degenerate" that works over all fields and rings -- rather than non-degenerate symmetric bilinear forms). The reason for considering such structures and guessing that they might even account for "nearly all" finite simple groups is *not* initially motivated by the idea that preserving such a form gives a finite group and one hopes that it is nearly simple (in the sense that passing to a Jordan-Holder series, or a very mild kind of composition series with the derived group and a central quotient thereof). Rather, it is due to the very specific way in which certain linear algebra constructions account for nearly the *entirety of the classification* of non-abelian simple Lie algebras and non-abelian simple Lie groups over $\mathbf{C}$ in a manner that is ultimately "defined over $\mathbf{Z}$". That suggests the possibility (originally due to Chevalley, building on much case-by-case work done by Dickson, Dieudonne, and others) to imitate such constructions over finite fields to get finite groups whose internal structure (e.g., simplicity) can be proved and studied in a unified manner.

The work of Borel and Tits (building on Chevalley's work over algebraically closed fields) provides a completely uniform structure theory for connected semisimple algebraic groups $G$ over *arbitrary* fields $k$ (not just finite fields), involving a classification expressed in terms of root systems and Galois cohomology (and making no mention whatsoever of linear algebra constructions or bilinear/quadratic forms). This gives rise to a general procedure for describing all possibilities for $G$ that are *absolutely simple* (in an algebraic group sense) in terms of a very specific array of algebraic structures: symplectic forms, non-degenerate hermitian forms, non-degenerate quadratic forms, Jordan algebras, octonion algebras, etc. The link to root systems also provides a systematic way to prove simplicity results for $G(k)$ or groups very near to $G(k)$ (e.g., quotient by center when $G$ is simply connected and $k$ isn't too small in a universal sense)

In the special case of finite fields (whose Galois cohomology is particularly simple, due to a combination of a result of Lang on torsors over finite fields and the pro-cyclicity of their absolute Galois group with "Frobenius" generator) some of these auxiliary algebraic structures only exist in a very limited manner (e.g., no non-trivial finite-dimensional central division algebras over such fields). Consequently, up to a very tiny list of exceptions, the exhaustive list of such groups $G(k)$ turns out to be explained by a very limited set of linear algebra constructions related to non-degenerate hermitian (= sesquilinear) forms, non-degenerate quadratic forms, and symplectic forms. And from knowledge of exceptional isogenies among such $G$'s one is led to the additional Ree and Suzuki groups in characteristics 2 and 3 by a very conceptual process (not how they were first discovered, but that's a separate matter).

So the reason one focuses largely on those specific constructions when studying "finite groups of Lie type" is a combination of (i) experience with Lie theory over $\mathbf{C}$, (ii) the fact that in the classification of root systems all but 5 are explained by specific linear algebra constructions, and (iii) the very simple nature of Galois theory for finite fields. (That the finite simple groups obtained in this way happen to account for nearly all finite simple groups is a real miracle.)