I believe the goal is to force the category of l-adic sheaves to keep some reasonable connection to geometry. No doubt someone more knowledgeable can say more about that, but I think if you start by understanding that torsion local systems are, by definition, representable by etale covers by finite relative group schemes, you will already realize why just translating the words into the full l-adic setting won't work. Namely, it is impossible to have an etale covering space that actually represents even the constant sheaf on the l-adic integers. Even at a lower level, you will find that a local system for $\mathbb{Z}/\ell^n \mathbb{Z}$ gets trivialized over progressively coarser covers as you reduce it mod powers of l, so you'd expect that an actual l-adic "local system" is only trivialized on the "intersection" of arbitrarily fine covers. This is not a cover; it is an inverse system of covers.

Why keep the connection to geometry? Again, not an expert, but I believe the biggest draw of etale cohomology is that it has a natural action of the etale fundamental group, which in the number-theoretic setting means some Galois group. For torsion local systems it's obvious that the fundamental group acts, since it is by definition constructed from automorphisms of finite etale covers, and so by actually building up an l-adic local system from torsion ones you retain the action. Furthermore, since the fundamental group itself has the inverse limit (profinite) topology, the representations you get are continuous.

Passing to fields: why tensor with $\mathbb{Q}_\ell$? Because first of all, this is what happens to module homomorphisms when you tensor with the fraction field. Why should we expect that any $\mathbb{Q}_\ell$ sheaf is like a $\mathbb{Z}_\ell$ sheaf tensored with the fraction field? Because, effectively, we have built finite generation into the construction, so we can always "find generators" with the largest possible denominators.

Finally, why take the limit over all finite extensions of $\mathbb{Q}_\ell$? Because again, by "finite generation" any "module" over the algebraic closure should be expected to have all its "structure coefficients" defined over some finite extension.

The construction is not at all ad-hoc; the idea is to start with a totally reasonable geometric definition and then apply standard algebraic tools (limits and colimits, though of categories rather than objects) to obtain a category still with good geometric properties but now also with good algebraic properties (defined over an algebraically closed field of characteristic zero).