I think the dual role of means (tool) and end in itself is part of the reason mathematics is so difficult to motivate. There is this constant tension within the pedagogy between the desires to teach the beauty and the practicality of mathematics.
From what I’ve seen, the beauty side tends to come from mathematics teachers and professors who have developed a passion for the subject and genuinely want to share that passion with their students. The practicality side, on the other hand, seems to be driven by industry, administrators, lawmakers, parents, and others involved in the curriculum-setting process.
Ultimately, both sides struggle mightily for students’ attention and a typical result is a collective sigh from the class. If were to make any progress I think we need to recognize that student motivation comes from within. Some students are genuinely attracted to the beauty of mathematics while others are interested in different subjects (science and engineering) for which mathematics is only a prerequisite. Perhaps one approach for dealing with this disparity might be to offer students a choice of projects with differing mathematical styles (proof vs calculation).