One thing to keep in mind is that the process that starts with LAGs and ends up with root systems is not "functorial". To keep everything very definite, let me restrict attention to finite-dimensional semisimple Lie algebras over $\mathbb C$. Then there are maps {isomorphism classes of s.s. complex Lie algebras} ↔ {isomorphism classes of Dynkin diagrams}, and at the level of isomorphism classes, the two maps are inverse to each other. In fact, there is a wonderfully functorial map going ←, i.e. from Dynkin diagrams to algebras, which was worked out by Serre, I think (maybe Chevalley). But the → map requires making all sorts of choices: pick a Cartan subalgebra, pick a notion of "positive" for it. Let $\mathfrak g$ be the Lie algebra and $\mathfrak h$ the chosen Cartan; then the group $\operatorname{Aut}(\mathfrak g)$ acts transitively on choices of simple system, and the stabilizer is precisely $\mathfrak h$. (Or, rather, in $\operatorname{Aut}(\mathfrak g)$ there are "inner" automorphism and "outer" automorphisms, and the inner ones in fact act transitively, and $\operatorname{Out}/\operatorname{Inn}$ acts as non-trivial diagram automorphisms. By "precisely $\mathfrak h$" I mean that the stabilizer is $\exp\, \mathfrak h \subset \operatorname{Inn}$.) So the space of choices is a homogeneous space for $\operatorname{Inn}(\mathfrak g)$ (which is the smallest group integrating $\mathfrak g$) modeled on $\operatorname{Inn}(\mathfrak g)/\exp(\mathfrak h)$. But, anyway, the point is that $\exp(\mathfrak h)$ acts nontrivially on $\mathfrak g$ but, as I've said, trivially on the Dynkin diagram, and hence trivially on the group that you construct from the Dynkin diagram.
This might be why you don't like the notion of root systems: you really do need to make choices to identify algebras with their root systems. It's something like picking a basis for a vector space — great for computations, but not very geometric. As a precise example: for $\mathfrak{sl}(V) = \{x\in \operatorname{End}(V) \text{ s.t. }\operatorname{tr}(x)=0\}$, a choice of root system is the same as a choice of (ordered) basis for the vector space $V$.
On the other hand, I claim that you should like the representation theory of a LAG. One way to study this representation theory (I might go so far as to say "the best way") is to pick a root system for your LAG and look at how $\mathfrak h$ acts, etc. Then, for example, finite-dimensional irreducible representations of a semisimple Lie algebra $\mathfrak g$ correspond bijectively with ways to label the Dynkin diagram with nonnegative integers. So you can really get your hands on the representation theory.
But representations of a group $G$ is a very geometric thing. There is some sort of "space", called "$BG$" or "$\{\text{pt}\}/G$", for which the representations of $G$ are the same as vector bundles on this space. If you don't like thinking about "one $G$th of a point", there are homotopy-theoretic models of $BG$.
You should also think of categories as geometric. Think about the case when $G$ is a (finite) abelian group. Then you might remember Pontrjagin duality: the irreducible representations of $G$ are the same as points in the dual group $G^\vee$. Then, at least for $G$ a semisimple LAG, you might think of its finite-dimensional representations as being like the points of some "space" $G^\vee$. The difference is that in the abelian case, all the points on $G^\vee$ correspond to one-dimensional representations, whereas in the semisimple nonabelian case in general the points are "bigger". The points are parameterized by the positive weight lattice, but they aren't actually the positive weight lattice. But the space "$G^\vee$" is some space noncanonically-isomorphic to the positive weight lattice. Again, this is like how the Euclidean plane is noncanonically isomorphic to the Cartesian plane.
Best Answer
A good way to handle systems of braid group representations is to consider the category of functors $\mathcal{C}\to R\textrm{-Mod}$, where $\mathcal{C}$ is a category with the braid groups as automorphisms. The braid groupoid $\beta$ (ie the groupoid with natural numbers as objects and braid groups as automorphisms) is then a subcategory of such $\mathcal{C}$. Note that $\beta$ itself is not quite satisfactory for such $\mathcal{C}$ since a functor $\beta\to R\textrm{-Mod}$ encodes a family of representations where the representations of $B_{n}$ is independent of the one of $B_{n+1}$. In other words, we would like $\mathcal{C}$ to encode compatibilities between the representations. There already exist good candidates for such category:
the partial braid category, see Section 2.3 of Palmer https://arxiv.org/pdf/1308.4397.pdf;
the Quillen’s bracket construction applied to $\beta$, denoted by $\mathcal{U}\beta$, and applied by Randal-Williams and Wahl https://arxiv.org/abs/1409.3541 [RWW].
The latter has the significant advantage that a large class of classical families of representations of the braid groups define functors $\mathcal{U}\beta\to R\textrm{-Mod}$:
the Burau representations; see Example 4.3 of [RWW].
the Tong-Yang-Ma and Lawrence-Krammer-Bigelow representations; see Section 1.2 of https://arxiv.org/pdf/1702.08279.pdf [S1].
the whole family of the Lawrence-Bigelow representations; see Section 5.2.1.1 of https://arxiv.org/pdf/1910.13423.pdf [PS].
Also, there are notions of polynomiality which allows us to characterise and prove more properties on these systems of representations:
the notion of (strong) polynomiality, a.k.a finite degree coefficient systems: the Burau representation is of degree $1$ (see Example 4.15 of [RWW]), the Tong-Yang-Ma representation is of degree $1$ and Lawrence-Krammer-Bigelow representations is of degree $2$ (see Propositions 3.25 and 3.33 of [S1]). This is the appropriate notion to prove twisted homological stability result see [RWW].
the notion of weak polynomial functors, which has originally been introduced for symmetric monoidal categories (for instance $FI$) by Djament and Vespa https://arxiv.org/abs/1308.4106, and generalised to categories of the same type as $\mathcal{U}\beta$ (namely pre-braided monoidal categories) in https://arxiv.org/pdf/1709.04278.pdf (see Section 4.2). An advantage of this notion is that it reflects more accurately than the strong polynomiality the behaviour of functors in the stable range. For instance, Church, Miller, Nagpal and Reinhold https://arxiv.org/pdf/1706.03845.pdf compute the weak polynomial degree (named "stable degree" in this paper) of some FI-modules. Moreover, denoting by $Pol_{d}(\mathcal{U}\beta)$ the category of weak polynomial functor of degree less or equal to $d$, we can define the quotient category
$$Pol_{d+1}(\mathcal{U}\beta)/Pol_{d}(\mathcal{U}\beta).$$
These quotient categories provide a new tool to handle families of representations with a sensible way to classify them. In particular, it doesn’t involve decomposition into irreducibles. See also Palmer https://arxiv.org/pdf/1712.06310.pdf for a comparison of the various instances of the notions of twisted coefficient system and polynomial functor. Hence weak polynomiality might be viewed as a refinement of representation stability phenomena and a sensible to talk about system of braid group representations.