It's used because, as your article states, "Despite the fact that the assumption of a flat Earth is fundamentally wrong, calculation of areas, angles and lengths using latitude and longitude can be complicated, so plane coordinates persist because they are convenient. The calculations can be done with plane trigonometry…"
The reason why they think or assert they are using it is rather irrelevant compared to the main point that they are using it.
Why would the reason for using something be irrelevant to using something?
Because the main point is that they are using them. The question of why it is in use compared to the statement that they are in use is a different line of inquiry entirely and does nothing to contradict it.
Where does it say in the article that "the idealized spherical world model is based upon those flat maps."? I can't find that anywhere.
Right here:
http://www.boshamlife.co.uk/wp-content/uploads/2013/11/PrimeMeridian.pdf “ By 1911, the Greenwich meridian had been accepted as the prime meridian for the whole world. However, relating the maps of an individual country or region to a standard system of latitude and longitude is not only difficult, it is nearly impossible.
The earth is approximately spherical, but maps are flat. They are fitted as closely as possible to the surface of the earth in one region, but when fitting them to a standard system of latitude and longitude, there are bound to be slight discrepancies. The differences between the coordinate systems used by different maps really didn’t matter until recently. When the GPS system was introduced in the 1980s, it was realised that having dozens of ‘local’ systems of latitude and longitude for different countries wasn’t going to work. A single coordinate system had to be devised, which would give the best results for every part of the world. It is known as
WGS 84 (World Geodetic System 1984). ”
The spherical earth is based on "flat maps".
You are wrong. The State Plane mapping system was developed in the 1930's. It uses two globe projections, both spherical, Lambert Conformal & Transverse Mercator, depending on the State shape.
Incorrect.
https://en.wikipedia.org/wiki/State_Plane_Coordinate_System “ The
State Plane Coordinate System (SPS or SPCS) is a set of 124 geographic zones or coordinate systems designed for specific regions of the United States. Each state contains one or more state plane zones, the boundaries of which usually follow county lines. There are 110 zones in the contiguous US, with 10 more in Alaska, 5 in Hawaii, and one for Puerto Rico and US Virgin Islands. The system is widely used for geographic data by state and local governments. Its popularity is due to at least two factors. First,
it uses a simple Cartesian coordinate system to specify locations rather than a more complex spherical coordinate system (the geographic coordinate system of latitude and longitude). By using the Cartesian coordinate system's simple XY coordinates, "plane surveying" methods can be used, speeding up and simplifying calculations. ”
It literally has the word "plane" in the name.
The North American Datum (NAD) is the horizontal datum now used to define the geodetic network in North America. A datum is a formal description of the shape of the Earth along with an "anchor" point for the coordinate system.
Lower down in the above article we read that the State Plane Coordinate Systems are associated with the North American Datum of 1983:
“ Originally, the state plane coordinate systems were based on the North American Datum of 1927 (NAD27). Later, the more accurate North American Datum of 1983 (NAD83) became the standard (a geodetic datum is the way a coordinate system is linked to the physical Earth). More recently there has been an effort to increase the accuracy of the NAD83 datum using technology that was not available in 1983. ”
The United States Government echoes the same association:
https://catalog.data.gov/dataset/united-states-stateplane-zones-nad83 “ United States Stateplane Zones - NAD83 Metadata Updated: August 11, 2016
U.S. State Plane Zones (NAD 1983) represents the State Plane Coordinate System (SPCS) Zones for the 1983 North American Datum within United States. ”
These systems involve flat coordinate systems. There may be a backend element which uses an ellipsoid to connect to other systems for converting coordinates between geographic models, but the data is flat. Utah's page
The Earth is Not Round! Utah, NAD83 and WebMercator Projections says that the spherical models are getting data from the flat ones.