First it fails, because pretty much everything Kant wrote about geometry runs into the serious problem that his whole idea is deeply connected to Euclidean geometry being the one, true correct geometry.

I don't recall him ever restricting himself to only Euclidean geometry. In Critique of Pure Reason, "geometry" is mentioned twenty times (each paragraph a separate quote; Markdown is being dumb):

Just as little is any principle of pure geometry analytical. "A straight line between two points is the shortest," is a synthetical proposition.

Thus, moreover, the principles of geometry--for example, that "in a triangle, two sides together are greater than the third," are never deduced from general conceptions of line and triangle, but from intuition, and this a priori, with apodeictic certainty.

Geometry is a science which determines the properties of space synthetically, and yet a priori. What, then, must be our representation of space, in order that such a cognition of it may be possible? It must be originally intuition, for from a mere conception, no propositions can be deduced which go out beyond the conception, and yet this happens in geometry.

Thus it is only by means of our explanation that the possibility of geometry, as a synthetical science a priori, becomes comprehensible.

Take, for example, the proposition: "Two straight lines cannot enclose a space, and with these alone no figure is possible," and try to deduce it from the conception of a straight line and the number two; or take the proposition: "It is possible to construct a figure with three straight lines," and endeavour, in like manner, to deduce it from the mere conception of a straight line and the number three. All your endeavours are in vain, and you find yourself forced to have recourse to intuition, as, in fact, geometry always does.

Geometry, nevertheless, advances steadily and securely in the province of pure a priori cognitions, without needing to ask from philosophy any certificate as to the pure and legitimate origin of its fundamental conception of space.

Footnote: Motion of an object in space does not belong to a pure science, consequently not to geometry; because, that a thing is movable cannot be known a priori, but only from experience.

On the other hand, the self-evident propositions as to the relation of numbers, are certainly synthetical but not universal, like those of geometry, and for this reason cannot be called axioms, but numerical formulae.

Empirical intuition is possible only through pure intuition (of space and time); consequently, what geometry affirms of the latter, is indisputably valid of the former.

But in this case, no a priori synthetical cognition of them could be possible, consequently not through pure conceptions of space and the science which determines these conceptions, that is to say, geometry, would itself be impossible.

But mathematics does not confine itself to the construction of quantities (quanta), as in the case of geometry; it occupies itself with pure quantity also (quantitas), as in the case of algebra, where complete abstraction is made of the properties of the object indicated by the conception of quantity.

Thus, when one quantity is to be divided by another, the signs which denote both are placed in the form peculiar to the operation of division; and thus algebra, by means of a symbolical construction of quantity, just as geometry, with its ostensive or geometrical construction (a construction of the objects themselves), arrives at results which discursive cognition cannot hope to reach by the aid of mere conceptions.

We shall, accordingly, show that the mathematical method is unattended in the sphere of philosophy by the least advantage--except, perhaps, that it more plainly exhibits its own inadequacy--that geometry and philosophy are two quite different things, although they go band in hand in hand in the field of natural science, and, consequently, that the procedure of the one can never be imitated by the other.

Of the two kinds of a priori synthetical propositions above mentioned, only those which are employed in philosophy can, according to the general mode of speech, bear this name; those of arithmetic or geometry would not be rightly so denominated.

For the assertion that the reality of such ideas is probable is as absurd as a proof of the probability of a proposition in geometry.

Other than this last quote (which is simply wrong), all of the other mentions consider geometry either as 1) a mere example or 2) in the context of phenomenal experience, which is predominately Euclidean for standard human beings on Earth. One could easily take it as a partial statement of the psychological unity of humankind.