For the growth model we can come up with ideas why the $\phi_1$ should not be equal to 1. Also for other models that deal with fractions it might be wrong as well. This is in particular problematic when the curves are fitted based on little data in a small range and subsequently extrapolated. This may contribute, for instance, to the piranha problem. Assuming linearity of effects on the odds can be problematic.

For the growth model we can come up with ideas why the $\phi_1$ should not be equal to 1. Also for other models that deal with fractions it might be wrong as well. This is in particular problematic when the curves are fitted based on little data in a small range and subsequently extrapolated. This may contribute, for instance, to the piranha problem. Assuming linearity/addition of multiple effects on the log-odds can be problematic.

More than semantics

The problem with the point of the book is that sometimes a (generalized) linear model can be a mechanistic model. The point is not about the idea whether the growth curve is a linear function or not (in some way or another), but about the idea that polynomials are often standard fitting methods without mechanistic meaning. The idea is that a non-polynomial function could be better. And whether this non-polynomial function is a GLM model that can still be interpreted as linear, or a non-linear model, that is besides the point.

So, for the sake of the point of the book, you can interpret 'linear function' in the narrow sense as a polynomial function, or as ordinary least squares regression. It is not about the semantics of what is or what is not a linear model.

Short

For the sake of the point of the book, you can interpret 'linear function' in the narrow sense as a polynomial function, or as ordinary least squares regression.

It is not about the semantics of what is or what is not a linear model. In this case the example is not a GLM (more about that below), but that is besides the point. Even if it would be GLM then it would be different from a polynomial function and the parameters $\phi_2$ and $\phi_3$ would have meaningful physical interpretations.

More than semantics

The problem with the point of the book is that sometimes a (generalized) linear model can be a mechanistic model. The point is not about the idea whether the growth curve is a linear function or not (in some way or another), but about the idea that polynomials are often standard fitting methods without mechanistic meaning. The idea is that a non-polynomial function could be better. And whether this non-polynomial function is a GLM model that can still be interpreted as linear, or a non-linear model, that is besides the point.

The problem with the point of the book is that sometimes a (generalized) linear model can be a mechanistic model. The point is not about the idea whether the growth curve is a linear function or not (in some way or another), but about the idea that polynomials are often standard fitting methods without mechanistic meaning. The idea is that a non-polynomial function could be better. And whether this non-polynomial function is a GLM model that can still be interpreted as linear, or a non-linear model, that is besides the point.

So, for the sake of the point of the book, you can interpret 'linear function' in the narrow sense as a polynomial function, or as ordinary least squares regression. It is not about the semantics of what is or what is not a linear model.