As someone who's been using actual GPS for geotagging my photographs, I have developed an intuition of just how much one can expect from it. I'd encourage you to do the same.
I have used an
actual GPS in my aircraft for descending to earth, at night and in cloud, with mountains around me. This is scary, but used by many privately owned aircraft every day, as well as the 'big' airlines.
The use of GPS, without any secondary source of navigation, is allowed by the national aviation authority and their reasoning for it is that:
GPS derived position accuracy is currently stated to be 100m or less, 95% of the time, and 300m or less, 99.9% of the time.So they have designed their routes and approaches to airports, descending through mountains, based on this criteria, with an additional margin obviously for instrumentation errors, piloting errors etc.
Their main interest is the safety of the general public, which we are all grateful for.
People attempted to determine their distance and they could not.
FWIW this makes your point arguing at the margins, as people (in planes) determine their distance from mountains every day by use of GPS and they all survive to do it again ... the next day.