> considering the coordinate system as spoiling the purity
Isn't that one of the main arguments for "coordinate free" or "synthetic" anything? It's about a pursuit of elegance and increased generality, not necessarily of obnoxious real-world politics; so it seems especially weird to inherently conflate these things.
Depends way you look at it. In some ways, analytic geometry generalises a set of algebraic tools to geometric problems. Eg from the text
> For the analysts, this was irrelevant: algebra captured the essential relations expressed by the terms of the problem, which then served to guide the mathematician toward the solution. For the synthetics, by contrast, a solution to the original geometrical problem could only be geometrical in nature; and so, what the analysts were offering were not solutions but meaningless numbers.
> While the analysts strove for maximum generality, the synthetics argued for the specificity and locality of all mathematical methods. […] [For the synthetics] even more misleading would be to believe that there is a single universal method that can be applied to all kinds of problems.
This does not mean that coordinate free geometry is reactionary per se, but considering the coordinate system as spoiling the purity of geometry is.