A single central dish would be cheaper, simpler, and easier to staff. It is not what we have, because on a globe a radar beam climbs above the weather, and then into space, long before it crosses the country.
A weather radar sends a signal outward in a nearly straight line and listens for the echo off rain and storms. On a flat Earth that signal would skim near the ground no matter how far it traveled, so one dish in the middle of the country could watch the weather everywhere. That would also be the cheaper and simpler choice by far: one radar to build, a small plot of land to hold it, and a single crew to run and repair it, in place of a nationwide fleet. No agency would pay for scores of costly radars, and the towers, power, and staff behind each one, if a single dish could do the work. So why is that not what we have today?
On a round Earth it cannot: the ground curves down and away beneath the straight signal, so the farther out it goes, the higher above the ground it ends up. Within about 143 miles the signal is already too high to catch storms near the ground, which is why the United States runs a grid of about 159 radars instead of one.
Left, the round Earth: KLOT (Chicago) reaches about 143 miles before its signal climbs too high to see storms near the ground, with a faint hint of the neighboring grid. Right, the flat-Earth claim: one dish covering the continent, with each city marked by how high the real round-Earth signal rides above it — 142 km over Denver is past the Karman line, in space. The two maps are drawn at very different scales. Height uses the standard 4/3 effective Earth radius (8,495 km) at the 0.5° lowest operating angle.
Start with the bill for what we have. The United States runs 159 of these radars. A single one carries a life-cycle cost near 38 million dollars over fifteen years, counting the tower, the electronics, the upgrades, and the crews who run and repair it. Across the network that is on the order of 6 billion dollars every fifteen years, roughly 400 million a year, plus a central operations center in Oklahoma and a backup generator and fuel tank at every site for the moments when the power fails. If one dish in the middle of the country could do the same work, that whole program would be waste. No agency spends billions on the hard way when an easy way exists.
Now ask what a single central dish would need if the ground were flat and the only limit were power. A weather radar detects rain, which fills the beam, so the returning signal fades with the square of the range. To reach a coast about 1,600 miles away, in place of its usual 143, a dish would need roughly 125 times the transmit power. A working radar here peaks at 750 kilowatts. That central dish would need on the order of 100 megawatts, in short bursts, from one antenna. The most powerful radars ever built peak near 15 megawatts. This is several times beyond anything ever made to work.
Power is only half the problem. A radar sees fine detail because its beam is narrow, and a beam stays narrow at great range only if the dish is huge. A working radar resolves weather to about two miles at 143 miles. At 1,600 miles that same dish blurs everything within twenty-five miles into one smear, far too coarse to find a tornado or pick out a single storm. To hold two-mile detail that far out, the dish would have to grow from 28 feet across to roughly 300 feet, and still swing a full circle every few minutes. No such antenna exists, and none is on the way.
A common retreat is to grant that one is not enough and ask for ten large radars instead. On a flat plane that might help, because more power reaches farther. On a round Earth it does not, because the wall is not power, it is the curve. However large a radar is, its beam still climbs above low storms near 143 miles, where the ground has fallen away beneath it. Ten radars, each watching a 143-mile circle, cover about 640,000 square miles. The lower 48 states cover about 3 million. So ten big radars would leave roughly four-fifths of the country blind to low weather, whatever their size. Tiling the whole area takes about fifty at a bare minimum, and closer to 159 once overlap and terrain are counted. The number is not set by budget or ambition. It is the map divided by the reach the curve allows. Make the Earth flat and the number falls. Keep it round and you need scores of them, which is what we have.
In a live debate someone will reach for an exotic radar and ask why that one does not just watch the whole country. Each has a clean answer, and the answer is always the same: every radar that does see past the horizon works only by using the round Earth. Not one is a single dish reaching across a flat plane.
The exotic radars do not poke a hole in the argument. Each one confirms it. Every path past the horizon runs through the round Earth: bounce off its upper shell, ride its curved conducting sea, or climb above it and look down. The one thing no radar can do is sit in the middle of a flat country and see it all. That is why there are 159.
Sources. Network size, range, and specifications: NOAA Radar Operations Center (roc.noaa.gov) and the WSR-88D reference. The beam climbing with range is stated by the National Weather Service (weather.gov). Beam heights use the standard 4/3 effective Earth radius, 8,495 km. Spaceborne precipitation radar: NASA GPM and TRMM (gpm.nasa.gov). Cost figures follow a NOAA life-cycle memorandum, about 38 million dollars over fifteen years per radar, and the 150 million dollar Service Life Extension Program (weather.gov). The power and antenna scalings are derived from the weather-radar equation, treating rain as a volume target, so the return falls as one over range squared.