At least for scalar equations $\dot x(t)=f(t,x(t))$, that is with a nonlinearity $f\in C^0(\Omega,\mathbb{R})$, defined on an open set  $\Omega\subset \mathbb{R}^2$, the Zorn's lemma is not necessary: the order structure of $\mathbb{R}$ allows to select a preferred solution (actually, two)

Any IVP  admits  an upper and a lower solution, whose domains are maximal intervals. For  $(t_0,x_0)\in\Omega$, define, for $t\in\mathbb{R}$
$$x ^ * (t):=\sup\{x(t)\\ : x\in C^1(\mathrm{co}(t_0,t),\\ \mathbb{R}),\\ \mathrm{graph}(x)\subset\Omega, x(t_0)=x_0, \dot x(s)=f(s,x(s))  \}\\ ,$$
(where of course $\sup\emptyset=-\infty$). Define
$\mathrm{dom}(x ^  *)$ to be the connected component of $t_0$ in the set $\{ t: x ^ *(t) > -\infty\}$. Then, $x ^ *$ is a solution of the ODE with IVC $x(t_0)=x_0$, maximally defined on the interval $\mathrm{dom}(x ^  *)$.