Can (some) linear regression model this (population) function accurately? James, Witten, Hastie, Tibshirani on page 35 of their book state with reference to the figure below:

In Figure 2.11, the true f [given by the black curve] is substantially
non-linear, so no matter how many training observations we are given,
it will not be possible to produce an accurate estimate using linear
regression.


Is this statement correct? Keeping in mind the widespread misunderstanding that linear regression can only model linear relationships/missing the fact that linear regression refers to linearity in parameters and can accommodate a large collection of non-linearities, I wonder whether this statement is correct. And if it is, I wonder what it is about this particular shape of f that puts it out of reach for the non-linearities that can be accommodated by linear regression?
page 35:

"... linear regression assumes that there is a linear relationship
between Y and $X_1$,$X_2$,...,$X_p$."

page 90:

"The linear regression model assumes that there is a straight-line
relationship between the predictors and the response."

On page 91 they call the model in 3.36 a linear model, even with an exclamation mark.
page 91:

(3.36) $mpg=\beta_0+\beta_1*horsepower+\beta_2*horsepower^2+\epsilon$
Equation 3.36 involves predicting mpg using a non-linear function of
horsepower. But it is still a linear model! That is, (3.36) is
simply a multiple linear regression model with $X_1=horsepower$ and
$X_2=horsepower^2$.

 A: A straight line is always going to leave something to be desired. In that sense, the claim is correct.
However, the true relationship looks cubic, which means that $\mathbb E[Y\vert X] = \beta_0 + \beta_1X + \beta_2X^2 + \beta_3X^3$ might be a reasonable model. Since this is linear in the parameters, the model is a linear regression, yet the fit should be reasonable. In that sense, the claim in incorrect.
Once you get into function convergence theorems in mathematical analysis, you will see that linear combinations of polynomial terms can get arbitrarily close to an awful lot of functions.
A: I wouldn't say the authors are wrong as such, but they weren't adequately careful with the wording. Often it's clear from context whether you mean simple or multiple linear regression, but here it wasn't.
The authors should have written

...it will not be possible to produce an accurate estimate using simple linear regression.

...where simple linear regression means using a single predictor: only $X_1$.
You are right that in this same paragraph, the authors had been discussing (multiple) linear regression in general:

For example, linear regression assumes that there is a linear relationship between $Y$ and $X_1$, $X_2$, ..., $X_p$.

There's no reason you couldn't define e.g. polynomials such as $X_2=X_1^2$ and $X_3=X_1^3$, in which case a linear model should indeed be able to fit that true $f$ reasonably well.
Edit: Alternately, to @whuber's point, they could have said

...it will not be possible to produce an accurate estimate using a regression that is linear in $X$.

Especially given that in Chapter 3 the authors note how a polynomial regression (nonlinear in $X$) is still a linear model (linear in the $\beta$s), it would have been helpful to be more explicit here. Their point here was not to say that the $f$ plotted in Figure 2.11 can't be well-approximated by a model linear in the $\beta$s---only that it can't be well-approximated by a model linear in $X$.
