this is an excellent question. I would prefer to offer a different interpretation however. My dissertation was in mathematics education, and I was interested in the history of the function concept in schooling. Klein was not the only one who espoused an interest in utilizing the function concept as the backbone of the school curriculum, and it has been claimed as such for more than a century now. If you look at the current New York State high school mathematics curriculum, it is called A Story of Functions, CCSS identify functions as backbone, etc. etc.
What is important however, is to pay closer attention to what calculus means for Klein in comparison to the nature of the discipline as it has subsequently developed in school and curricula. Klein was German, and valued a Leibnizian view of calculus. There is a difference here, and approximation methods play a more pronounced role in this vision. If you notice, Klein later demands a more prominent role for "The Finite Calculus", this includes things like finite differences and interpolation; ideas which never made their way into the standard curriculum.
I think Klein's demands for a more humanistic and approximate style to the calculus resonates louder today with the rise of computing and statistics in culture, society, and mathematics.
While you can find something like a recursive sequence as a function in the CCSS, curricula and teaching still have a long way to go to valuing the finite and approximate. Too often we see these as specialized topics for later study rather than ways to fill out existing concepts like linear functions that are such mainstays in the curriculum. Some examples do exist however, maybe you'd be interested in the COMAP Mathematical Modeling Our World materials, or some older textbooks made for North Carolina high schools before the CCSS when they were teaching vectors, matrices, and discrete mathematics topics in classes like Algebra I.
So, yes, functions should be the backbone -- but a function is not a function is not a function. Functions have multiple interpretations through history, and we should aim for a less rigid notion where we study relationships in a variety of contexts as opposed to questions around function theory like "is this a function".
Yes, calculus should be the end game -- but calculus can be much more than what is presented in the AP Curriculum or MIT's first course. People arguing for statistics to replace calculus ignore the fact that statistics is just an application of calculus, particularly when you value topics like finite calculus and the probability calculus in an introductory calc setting.