A geometric interpretation of the envelope of a one-parameter family of surfaces is the following (slightly adapted from Wikipedia): a point on the envelope is a point in the intersection of two "adjacent" surfaces. If you don't mind infinitesimals this could be rephrased as: $(x_0,y_0,z_0)$ is a point on the envelope, if $(x_0,y_0,z_0)\in S_{a_0}\cap S_{a_0+da}$ (where $S_a$ denotes the surface and 0-subscripts denote values of the variables $x,y,z,a$.). But this condition is equivalent to the following: The values $(x_0,y_0,z_0)$ satisfy the equation $z=w(x,y;a)$ for a certain value $a_0$, and moreover, as we change $a$ slightly, keeping $x=x_0, y=y_0$ fixed, the value of $z$ does not change to first order. This is the condition $\frac{\partial w(x,y;a)}{\partial a}=0$.

I'm not sure what it means in mathematics for something to be formal or not. So I assume you are just expressing your discomfort with differentials $dz$, $dx$ etc. Unfortunately calculus books tend to discard them as "notation without meaning". The current mainstream approach to give them meaning is to interpret them as differential forms in the sense of differential geometry. I suggest you look at any book on the topic. You would then interpret the variables $x,y,z$ as maps $x:\mathbb{R}^3\to \mathbb{R}$ etc. (And you will also be confronted with the cotangent space.)

Since they are talking about a one parameter family of flat planes through a point, it should be geometrically clear, that the intersection of two adjacent planes is a line through $(x_0,y_0,z_0)$, hence the envelope is a union of such lines and is hence a cone.

A geometric interpretation of the envelope of a one-parameter family of surfaces is the following (slightly adapted from Wikipedia): a point on the envelope is a point in the intersection of two "adjacent" surfaces. If you don't mind infinitesimals this could be rephrased as: $(x_0,y_0,z_0)$ is a point on the envelope, if $(x_0,y_0,z_0)\in S_{a_0}\cap S_{a_0+da}$ (where $S_a$ denotes the surface and 0-subscripts denote values of the variables $x,y,z,a$.). But this condition is equivalent to the following: The values $(x_0,y_0,z_0)$ satisfy the equation $z=w(x,y;a)$ for a certain value $a_0$, and moreover, as we change $a$ slightly, keeping $x=x_0, y=y_0$ fixed, the value of $z$ does not change to first order. This is the condition $\frac{\partial w(x,y;a)}{\partial a}=0$.

I'm not sure what it means in mathematics for something to be formal or not. So I assume you are just expressing your discomfort with differentials $dz$, $dx$ etc. Unfortunately calculus books tend to discard them as "notation without meaning". The current mainstream approach to give them meaning, is to interpret them as differential forms in the sense of differential geometry. I suggest you look at any book on the topic. You would then interpret the variables $x,y,z$ as maps $x:\mathbb{R}^3\to \mathbb{R}$ etc. (And you will also be confronted with the cotangent space.) From this point of view the equations $dz=w_xdx+w_ydy$ and $0=w_{ax}dx+w_{ay}dy$ can be seen as the result of applying the de Rham differential to both sides of the equations $z=w(x,y;a)$ and $0=w_a(x,y;a)$, which (for fixed value of $a$) are just the defining equations of $\gamma_a$. Interpreting these differential forms as fields of covectors on $\mathbb{R}^3$, the system of these two equations can be seen as determining a field of one dimensional subspaces of the tangent space (the kernel of the covectors.). This then amounts to the same as your point of view, of parametrizing the curve $\gamma_a$ with a parameter $t$, and writing down the equations you wrote.

Since they are talking about a one parameter family of flat planes through a point, it should be geometrically clear, that the intersection of two adjacent planes is a line through $(x_0,y_0,z_0)$, hence the envelope is a union of such lines and is hence a cone.