No one can truly know what shape a country is as a whole.
Don't see why not. If you have a large database of feature locations where you know the latitude and longitude of each one and know what type of feature it is (road, river, coastline) and which territory it is part of (US state, country etc.) you can plot these on a globe and see the shape for yourself. Or you can just trust that others have done this already and buy yourself a globe or use Google Earth/Google maps etc.
That means you are projecting a cylindrical view on top of a sphere.
Not accurate.
Well not exactly. The feature locations are just expressed as pairs of numbers, latitude and longitude. There's no shape really, just pairs of numbers. You can choose to plot those on a flat surface or on a globe or on some other odd shaped surface if you like. You can project a sphere onto a cylinder and since that is a reversible process, you can certainly project a cylinder onto a sphere and you can always unwrap a cylinder to a flat surface. You can do the same with a cone instead of a cylinder if you want to.
However you do it, this process is generally known as projection and all the professional maps I've ever seen are some form of projection from a globe. Sometimes the name of the projection is quoted on the map, sometimes it isn't, but you can often go to the publisher and find the information from there. You could argue which came first the chicken or the egg - i.e. do we project from a flat map to a globe or a globe to a flat map, but to me, the fact that if you take any flat map and project it back to a globe using the correct reverse projection, you end up with the same globe you would from any other map with a completely different projection says that the globe is the correct representation. There is only one globe representation, but many flat maps with very different layouts.
Well if you are talking about large scale flat maps covering small areas, then nobody is disagreeing with you, but "all flat maps" necessarily include maps of the whole earth and there are lots of these, so for example Mercator, Mollweide, Winkel tripel, and that old favorite, the Azimulthal Equidistant. They all look very different, and contradictory, so they can't all be accurate, yet they are all flat maps.
Getting back to my point of the facetious demand of FE producing an accurate map.
RE cannot seem to come to a consensus of what it is they truly want.
Oh I think that's quite simple. If I want a true representation, I have my globe, but a globe large enough for me to use on a day trip somewhere is utterly impractical, so a large scale map will do just fine, since I live in the UK, if I want a long walk, I might take an Ordnance Survey map which uses a Transverse Mercator projection. On the other hand, if I want to make a cross country flight, I'll take a UK VFR Chart which uses a Lambert Conical Projection because this is a projection more suitable for aviation flight planning. Each type of map has it's own set of limitations and they vary from map to map, so it's a case of matching your needs to the pros and cons of each type of map and choosing the most appropriate for the task.
Where RE and FE adherents differ is that if you believe in RE, you know that a universal, accurate complete flat map of the earth is impossible, whereas if the earth is truly flat, this is not only possible. but it should be borderline trivial to produce. Yet where is it?