$\beta\omega$ is sequentially discrete, that is, every convergent sequence is eventually constant. Thus, since the space is not discrete, the space is not sequential, that is, it's untrue that every limit point of every set is witnessed by a sequence from the set converging to it.
Is $\beta\omega$ radially discrete, that is, every convergent transfinite sequence is eventually constant? If not, is the space still radial, that is, every limit point of every set is witnessed by a transfinite sequence from the set converging to it?
So $\beta\omega$ can be quickly seen to not be radial, or even pseudoradial: $\omega$ is a radially closed (since it is countable and sequentially closed) subset which is not closed.
However, it is consistently not radially discrete: under $CH$ each $P$-point has a local ordered basis of size $\omega_1$, so each $P$-point has an $\omega_1$-sequence of $P$-points converging to it.