We only see the carrier because it has such narrow bandwidth, the modulated data is for us not distinguishable from the noise. FYI, at this data rate the carrier contains 25% of the transmitted power.
The "live" plot (with the peak) uses indeed 1Hz bins (of 1 second), that we average over the last 2-3 minutes to reduce the noise. We could go even narrower, I might give that a try on the recorded data.