Let $\pi:P\rightarrow M$ denote a principal $G$-bundle, where $M$ is thought of as some spacetime and $G$ is an appropriate group (such as $\mathrm{U}(1)$ or $\mathrm{SU}(2)$).
I want to understand better why a connection form $A\in\Omega^1(P,\mathfrak{g})$ on $P$ is used to describe the physical notion of a gauge field.
What I have thought up until now: Via local section/ gauges $s_i:U_i\rightarrow P$ we call pull back the connection $A$ to local connection forms on $M$: $$A_i:=s_i^\ast A\in\Omega^1(U,\mathfrak{g}).$$ If $U_i\cap U_j\neq\emptyset$, then there is a transition function $g_{ij}:U_i\cap U_j\rightarrow G$ of the bundle such that $s_i(x)=s_j(x)\cdot g_{ij}(x)$ and $A_i$ transforms as $$A_i=\mathrm{Ad}(g_{ij}^{-1})A_j+\mu_{ij},$$ where $\mu_{ij}=g_{ij}^\ast\mu_G$ is the pulled back Maurer-Cartan form of $G$. By assuming that $G$ is a matrix Lie group, this transformation behavior is exactly that of a gauge transformation in physics.
Then it is said that we can "identify" this connection form with a gauge field.
If we do so, then the formulas for the local pulled back forms of the corresponding curvature form $F$ of $A$ look exactly like the formulas known from physics for calculating the field strength from the gauge potential,
$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu-[A_\mu, A_\nu],$$
(on an open set $U\subset M$ that is both trivializing for $P$ and a coordinate neighborhood for $M$), and the local forms $F_i:=s_i^\ast F$ transform as $$F_i=\mathrm{Ad}(g_{ij}^{-1})F_j,$$ which is also similar to what we see in physics. This seems like another good reason to think of a gauge field as a connection.
My $\textbf{question}$ now is: Are these transformation behaviors of the local forms of $A$ and $F$ the $\textit{only}$ reasons why we mathematically describe gauge fields by connections or are there some other results (physically or mathematically) that lead to this conclusion?
