I'm not an engineer, but it seems like having a single pivot at the bottom of a large mast would be favourable -- it'd allow the mast to freely flex in winds while being constrained by the support cables without causing damage, whereas a small base might just crumple if flexed too far.

A rigid base would have to be extremely strong to supply the promised rigidity, small base means big force to supply the turning moment. You can think in terms of it being a Dirichlet (fixed but flexible) vs. Cauchy (fixed and unflexible) boundary condition. Not to mention it being unnecessary; the only advantage I can think of is that it might exclude some undesirable oscillation modes, which is probably easier to achieve by attaching support cables at various points along its length.

That is to allow the tower to be flexible to wind induced vibration. If it would have been a standard three legged foundation then the constant vibration would have introduced cracks leading to unnecessary stiffening.