That seems to be a misunderstanding. The field-theoretic case behaves in essentially the same way as the point mechanical situation.

For instance, the arguments of the Lagrangian density $${\cal L}(\phi,\partial_t\phi,\partial_x\phi,\partial_y\phi,\partial_z\phi,t,x,y,z)$$ are independent variables in the same way as the arguments of the Lagrangian $$L(q,\dot{q},t)$$ are independent variables.

The proof is a straightforward generalization of this Phys.SE post.

That seems to be a misunderstanding. The field-theoretic case behaves in essentially the same way as the point mechanical situation.

1. On one hand, the arguments of the Lagrangian density $${\cal L}(\phi,\partial_t\phi,\partial_x\phi,\partial_y\phi,\partial_z\phi,t,x,y,z)$$ are independent variables in the same way as the arguments of the Lagrangian $$L(q,\dot{q},t)$$ are independent variables.

2. On the other hand, inside the actions $S[\phi]$ (and $S[q]$), the derivatives $\partial_{\mu}\phi$ (and $\dot{q}$) depend on $\phi$ (and $q$), respectively.

The proof is a straightforward generalization of my Phys.SE answer here.

That seems to be a misunderstanding. The field-theoretic case behaves in the same way as the point mechanical situation: The arguments of the Lagrangian density $${\cal L}(\phi,\partial_t\phi,\partial_x\phi,\partial_y\phi,\partial_z\phi,t,x,y,z)$$ are independent variables in the same way as the arguments of the Lagrangian $$L(q,\dot{q},t)$$ are independent variables. The proof is a straightforward generalization of this Phys.SE post.

That seems to be a misunderstanding. The field-theoretic case behaves in essentially the same way as the point mechanical situation.

For instance, the arguments of the Lagrangian density $${\cal L}(\phi,\partial_t\phi,\partial_x\phi,\partial_y\phi,\partial_z\phi,t,x,y,z)$$ are independent variables in the same way as the arguments of the Lagrangian $$L(q,\dot{q},t)$$ are independent variables. 

The proof is a straightforward generalization of this Phys.SE post.