I am not sure where or why this thread is trying to illustrate
The OP was trying to understand why, even though the radius of the Earth is smaller at the poles, one minute of arc along a meridian is longer. I believe very early on it was explained that the length of arc is dependent on the radius of curvature and not the radius of the Earth. The use of the term "nautical mile" has caused much of the following discussion.
The "international nautical mile" was fixed at 1852m many years ago. But even after that both the US and the British continued with their own defintions. Prior to that there were many definitions, all based on countries own navigation needs, and the datums they had adopted for charting. But all were based on an approximation of one minute of arc of a meridian. The basis for this measurement is based on charting. If you travel due north and gained one minute of latitude in one hour, then you had travelled one nautical mile.
The numerical value used for a nautical mile for critical procedure design is the value used, regardless of theory about the shape of the Earth.
And, the distances involved are small from a global (bad pun) standpoint.
Now we have a fixed definition of the "internation nautical mile" we have to work back the other way. To ignore the shape of the Earth is to accept an error in any co-ordinates calculated. After all the international definition is only an "average" value. To work out the change in latitude we need to know how it changes with latitiude. This may not be relevant for local area work. But the cumulative error on a navigation system can be quite large if not taken into account. It is also necessary to take this effect into account when locating items based on radar, if you wish to use any for of co-ordinate system other than one based on a range and bearing from yourself.
Ultimately there are many definitions of a "mile" all of which are convenient for their own calculations, including geographic mile, tactical mile, statute mile, and the list goes on.
As for using an FMS, someone has already (hopefully) calculated all these variations.