The only Starlink satellites that can hope to reach the north pole are the satellites that reach 82.4 degrees north/south (orbits at inclination of 97.6 degrees) and there's currently only 187 of those operating so very few will be in view at any one time and the Starlink dish needs to pick an orientation so you will only see some of them at any given time (FOV is only part of the direction it's facing, missing at least the other half of the satellites, if not at least 2/3 of the satellites in view).
Whether that makes service impossible, I'm not sure. US government may work with SpaceX to make a customized 360 degree antenna that can reach them. Or they may mount it on a tower. I think a customized antenna would be needed anyway to survive the temperatures there. Starlink is technically only rated down to -30C though people do use it below that temperature and it seems to work. I doubt it'd work at those -60C temps however.
Whether that makes service impossible, I'm not sure. US government may work with SpaceX to make a customized 360 degree antenna that can reach them. Or they may mount it on a tower. I think a customized antenna would be needed anyway to survive the temperatures there. Starlink is technically only rated down to -30C though people do use it below that temperature and it seems to work. I doubt it'd work at those -60C temps however.