EL equations with unbounded integrand at the boundary conditions - Divergent integral after change of variables as a more general problem

Consider an action integral $$I = \int_{t_0}^{t_f}L(\eta(\theta(t)),\dot{\eta}(\theta(t)),t)\,dt\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(0)$$ with boundary conditions $$\dot{\theta}(t_0) = \dot{\theta}(t_f)=0$$ for some strictly-monotonic quantity $$\theta(t)$$ satisfying $$\dot{\theta}(t) > 0$$ in the open interval $$t_0 < t < t_f$$.

Suppose, I want to find $$\eta(\theta)$$ minimizing $$I$$. To this end, I'm rewriting the action integral as

$$I = \int_{\theta_0}^{\theta_f} \tilde{L}(\eta(\theta),\eta'(\theta),\theta) d\theta = \int_{\theta_0}^{\theta_f} \dfrac{L(\eta(\theta),\eta'(\theta)\omega(\theta)(\theta),\theta)}{\omega(\theta)}d\theta\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)$$

with $$\omega(\theta) = \dot{\theta}(t)$$, $$\theta_0 = \theta(t_0)$$, $$\theta_f = \theta(t_f)$$, and $$\dot{\eta}(\theta)=\eta'(\theta)\dot{\theta}$$. Applying EL equations defines the boundary-value problem

$$\dfrac{d}{d \theta} \dfrac{\partial \tilde{L}}{\partial \eta'} - \dfrac{\partial \tilde{L}}{\partial \eta} = 0$$ with prime superscript notation defining the partial derivative with respect to the new variable of integration as $$q' = \partial q/\partial \theta$$ for some $$q$$ and BCs $$\eta(\theta_0) = \eta_0$$ and $$\eta(\theta_f) = \eta_f$$. Notice the difference in prime and dot notations with $$\dot{q}(\theta(t))$$ being time derivative of $$q$$ which can be defined as $$\dot{q} = q' \omega$$ with $$q' = \partial q/\partial \theta$$.

I am stuck at this final EL and how to approach it as it's unbounded at the BCs due to the presence of $$\omega$$ term at the denominator of Eqn (1) with $$\omega(\theta_0)=\omega(\theta_f)=0$$ despite the fact that the original Lagrangian $$L$$ is defined everywhere. How are we supposed to approach this problem regarding both steps of analytical derivation and implementation details to get rid of unboundedness?

Intuitive analogy to the problem : One can consider $$\theta(t)$$ as a generalized coordinate of a single DoF system. A simple analogy toward $$\eta$$, on the other hand, can be a Jacobian or general transmission ratio, which is a function of position. The system executes a point to point motion between two points $$\theta_0$$ and $$\theta_f$$. Due to bijective nature of the relation between $$\theta$$ and $$t$$, all functions of $$t$$ have a trivial realization in $$\theta$$ domain. In this regard, to avoid a notation abuse in the post, I do that by introducing $$\omega(\theta)$$ in Eqn. (1) which maps to velocities from positions instead of $$\dot{\theta}$$ which equivalently maps to velocities from time