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This answers a different question. When I mentioned algebra and calculus, I was talking about barriers to get through, not about a deeper sort of learning and interest.
In any event I whole heardly agree. There is something about geometry that leads much more quickly and easily into very abstract stuff.
But this path has a practical difficulty in the sense that a lot of advanced math isn't particularly useful in the likely applications found in most sciences. Sure geometry is at the heart of mechanics for example, but getting through most elementary physics requires calculus where the metric form of geometry dominates. The deeper spatial concepts are obscurded with a lot of calculation. You only get to see the geometric ideas you've using, when they are made more explicit in something like theoretical physics or cosmology.
One interesting example of how elementary geometry lead directly to core spatial constructs can be found by playing with dihedral group symmetries of the square. These are isomorphic to ridgit motions in euclidean vector spaces. There are also applications in chemistry that I don't know and a lot of other physical sciences.
On the other hand, playing with blocks, patterns, and geometrical arrangements is great way to develop this sort of thinking at an early age and certainly should be part of math-sci elementary education curriculum.
I think the reason such tools are not used in schools with any systematic understanding is simple. Most teacher have never been exposed to this sort mathematics and so they just don't know about it. Even if one or two might, other teachers and the school system would probably discourage it, since it looks like idle play. Cutting and folding paper, moving funny looking blocks around, permuting patterns, playing with patterns as tiles, making things out of them. That's seen as arts and crafts---stuff to do on a rainy day.
Getting late for work again..
CG