Okay I think I made a lot of implicit assumptions because I was confused by your functions in the last answer, which made my answer very confusing itself. So I try to be a bit more explicit this time. I will only consider your first function.
Now if this is a production function, then the $x_i$ are inputs, and you only have one output.
Now your production function has the following properties: $\frac{df}{dx_i}=\beta_i$ and $\frac{df}{dx_jdx_i}=0$. This implies that your inputs are completely independent from each other. You can reach the same output with only $x_1$ or with only $x_2$. This makes no sense if your inputs are Capital and Labor (at least not until we have fully automized production where you don't need a human anywhere in the production process). Because if either of them are 0 in the real world, you would get no output.
This consideration lead me to believe that your $x_1$ model methods of production, which implies that they already contain mixes of Capital and Labor. In that case you would simply optimize over these different methods of production, and pick the best one. The one with the highest $\beta_i$ to cost ratio. Because at a company level, marginal costs would most likely be constant.
This is a reasonable model, from a companies perspective. You can only choose between those production methods, so the fact that you mashed together Capital and Labor does not concern you. You optimize the function and pick the best method of production given fixed costs. The (presumed) simplification is, that you don't consider how expansion or production changes the marginal cost within a production method.
(If the expansion would be on a economic scale, then you would increase the wage of workers by employing more of them, which would at some point result in you picking a more capital heavy production method)
An economists approaches this from a different angle: They are interested in the Capital/Labor ratio and not in the specific method of production. They want a function that takes Capital and Labor as input, selects the best method of production given that input, and returns an output. Their assumption is, that on this scale there are so many different methods of production, that you can brush over them and basically get a continuous function in Capital and Labor.
They want a model which has the property $\frac{df}{dx_jdx_i}>0$ which the Cobb-Douglas function provides.
You are basically comparing particle simulation with fluid simulation. The equations to model a single water molecule will be different from modelling a stream of water. And it might look like one doesn't have anything to do with the other.
The other possibility I thought of, was that this function is actually a cost function of producing the outputs $(x_1, ...,x_n)$ but then you have a fixed marginal cost of $x_i$ by definition. Which again, is an assumption that is reasonable to make on the micro level but not on the macro level.