Ahh not quite though. There are really two discs in play:
1- the disc to measure total sunlight into Earth, this being the shadow the earth would generate
2- the implied mathematical equivalent disc that happens from geometrically flattening the spherical earth into a disc
#1 is simply correct as far as I can tell. That is how one would measure total sunlight in, and it does ‘spread out’ as you get to the poles, as the link says. If you could get more sunlight in by varying the geometry of an object then solar panels would be wavy not flat
#2 is where the quantumville flat Earth comes in. Because they equate this total energy in to energy out. They are saying that instantaneously, every speck of energy that comes in from the sun is instantaneously divided and radiated out across the entire surface area of the sphere, which mathematically comes out to 1/4th the energy out per square meter.
As energy in = energy out then also therefore (obviously; “trivially” ) this energy out of 1/4th sun energy per square meter now also becomes the energy in per square meter too:
[source]
Note well that “Incoming Solar Radiation” is shown as 342 W m-2, not 1368 W m-2.
All this mathemagics has the effect of treating the Earth as a flat 4x-larger disk where each m^2 of it receives (and emits) this 1/4th solar energy sunlight.