Pure peak power doesn't matter, but 5, 10, 20 sec power much more.
Yes, it's trainable and considering you're a newbie on track/sprint thing, and brought pure strength up as described, your peak, and 5, 10 sec, should come up considerably.
I can't get any reason for you not seeing improvements on short period power, but sometimes if you aren't fresh, peak power doesn't show at all.
What about time on real track for short efforts like first lap from standing starts, and flying 100m time and max speed?
I'm addicted to data (powermeters on all bikes since 2006, including track bikes) but the real gauge is the track.