I think the best way to think about this is in terms of operads.

Braided monoidal categories are representations of an operad $\Pi$ in the category of (small) categories.

The category $\Pi(n)$ has objects parenthesised permutations of $\{1,\ldots,n\}$ like $(4(23))1$. The morphisms $(\sigma) \to (\tau)$ are braids $\beta \in B_n$ so that $\beta$ maps to $\tau \sigma^{-1}$ under $B_n\to S_n$.

The operad structure
$$
\Pi(n) \times \Pi(k_1) \times \ldots \times \Pi(k_n) \to \Pi(k_1+\ldots+k_n)
$$
is given by replacing the $i$-th sting by the braid on $k_i$-strings.

Now any representations $\Pi(n) \to \underline{Hom}(\mathcal{C}^n,\mathcal{C}) = End(\mathcal{C})(n)$ induces a braided monoïdal structure with

- $\otimes$ corresponding to the object $(12) \in \Pi(2)$
- the associativity morphism corresponding to the trivial braid $(12)3 \to 1(23)$ in $\Pi(3)$
- the braiding corresponding to the morphism $(12) \to (21)$ in $\Pi(2)$ induced by the generator of $B_2 = \mathbb{Z}$.

MacLane's coherence theorem tells you that this is an equivalence.

All the operations you're looking at come from this operad structure.

The $\Pi(n)$ have a nice geometric interpretation. They are the fundamental groupoids of the operad of little discs $C_2(n)$ (or equivalently of $F(\mathbb{C},n)$ the spaces of configurations of $n$ points in the plane) restricted to a suitable collection of basepoints. This generalizes the classical definition $P_n = \pi_1(F(\mathbb{C},n),p)$, $B_n = \pi_1(F(\mathbb{C},n)/S_n,\overline{p})$.

I think this makes the whole picture a lot clearer because we get all of these operations as part of the same structure and we get a universal characterisation of
that structure and a geometrical interpretation for it.

This leads to other nice considerations. For example the Grothendieck-Teichmuller group $GT$ defined by Drinfeld is the automorphism group of the (prounipotent completion) operad $\Pi$. This explains why $GT$ is universal for quasi triangular quasi Hopf algebras as their representations form braided monoïdal categories and related to the Galois group of $\mathbb{Q}$ as $GT$ appears as an automorphism group of fundamental groupoïds of the algebraic varieties $F(\mathbb{A}^1_{\mathbb{Q}},n)$.