This seems almost exactly backwards to me. “Geometric” proofs (i.e. based on spatial reasoning) are fast and fluid, relying on an imagined spatial configuratio

This seems almost exactly backwards to me. “Geometric” proofs (i.e. based on spatial reasoning) are fast and fluid, relying on an imagined spatial configuration which can be seen all at once. “Algebraic” proofs (i.e. based on symbol manipulation) are serialized and low-bandwidth, and working them out takes a significant amount of laborious fiddling with symbols on paper, and is almost impossible to work out mentally except in nearly trivial cases. The benefit of the “algebraic” version is often that putting in that labor can often yield a result even when the prover has no special insight. That is, the method can be more reliably carried out by non-geniuses (assuming that the prover has sufficient patience and stamina), because the work can be broken into small individually opaque steps and written down on paper instead of needing to be seen all at once. To the extent algebraic proofs are fast/elegant/obvious, they generally rely on a clear 2-dimensional notation on paper where it is easy to see how to simplify the parts based on visual patterns known via extensive past experience. Cf. https://arxiv.org/abs/math/9205211 https://arxiv.org/abs/math/9205211