Summarizing comments as an answer.

The standard interpretation of radial does not require that $\alpha\mapsto a_\alpha$ be continuous. Therefore, the Fortissimo space is radial, despite the lack of non-trivial continuous maps from limit ordinals into the space. This also aligns with the standard result that every LOTS is radial: when taking the necessary well-ordered initial/cofinal transfinite sequence in $A\cap(\leftarrow,x)$ or $A\cap(x,\rightarrow)$ in KP's proof in the comments, there's no guarantee that this sequence will be closed in the space.

On the other hand, there does not seem to be standard terminology for the strengthening of radial that requires the transfinite sequences be continuous maps from limit ordinals.