The short-term stability comes from the quartz oscillator, not from the GNSS. This is nicely visible in the Allan deviation plot of the SRS FS740 GPS reference: https://www.thinksrs.com/images/instr/fs740/FS740_lockedLG.g...

Even the simplest TCXO is more stable than the GNSS up to a few ten s. The OCXO has the crossover at a few hundred s, and the Rubidium atomic clock at over one thousand s.

At these time scales, the jitter of the 1PPS measurement averages out very well and should not be a serious limitation.