Also, as defined by the French academy of science ~1800, the meter is 1/10,000,000th the distance from the north pole to the equator, on the meridian through Paris.

(The second is in turn precisely defined by particle physics.)

If that justifies knot then it also justifies kilometers, because it too was originally standardized on the earth's size.

Neither are exactly the size as originally defined.

A modern knot is 1852 meters. One minute of latitude is "about 1,855.325 m on the WGS 84 ellipsoid" says https://en.wikipedia.org/wiki/Nautical_mile. The difference is 0.16%.

A quarter of the Earth is:

  1855.325m * 60 seconds/degree * 360 degrees/circle /4 = 10018755.0
  = ~10018 km.

Straight lines on a Mercator projection aren't great circles, they're rhumb lines, and if anything they're a spiral on the sphere, not straight.

It's not the 1800's any more. We have computers and stuff, we don't have to rely on techniques developed for a time when your navigational tools consisted of a map, a compass and a sextant.

Note: The actual deviation from a great circle route is generally fairly minimal in practice.

PS: GPS is just another system that really can fail.

a Mercator projection will give you a bearing to steer to arrive at your destination, which can be followed by eye, autopilot, etc etc.

Calculating a great circle arc and plotting it, following a bearing which may only change by a degree or two but isn't constant, is an unnecessary complication on short routes.

Even commercial vessels will use those old techniques if they don't need to consider the longer range effects. Error correction in the human process is always a consideration.

(Not even talking about data entry errors in GPS systems, but that Malaysia - Melbourne story was Funny.)