Actually, no. FR can, and indeed is, calculated from IR, but you can't possibly conclude anything about FR by looking at step response. This almost perfect step response actually means that phase is linear.
Take, for example, step response of
Performa3 F208:
View attachment 27615
As you can see, not only that it is nowhere as perfect as @mitcho 's but it is totally time unaligned. First peak is from tweeter, second from midwoofer and third from woofer(s). However, this speaker has very linear FR and is considered to be one of the best (if not the best) speaker in it's price range.
So, the answer to your question is this: IR response contains both, frequency and phase information so everything is derived from it - frequency resposne, phase responsee, step response, GD, distortion, .. But looking at step response you can't conclude anything about FR. As step response is happening in time domain you can conclude only about phase (time coherence) of the speaker drivers. Let's also not forget that, as mighty as step response looks, it's significance on SQ is actually small to none. It is the FR that matters, esepecially when measured not only on-axis but on a wide range of horizontal and vertical angles, as is shown on famous spinorama graphs.
But, yes,
@mitchco speakers indeed have extremely linear FR.