That is because you’re expecting ground access points to magically communicate with a satellite at 0 degree elevation. With a realistic elevation to deal with o

That is because you’re expecting ground access points to magically communicate with a satellite at 0 degree elevation. With a realistic elevation to deal with obstacles, diffraction, and electronic steering limitations, you’ll end up with a radius of coverage closer to 200 miles, which would give you that factor of 100. This shorter radius also limits losses due to distance attenuation. Your naive calculation assumes the limiting factor is the curvature of the only planet, which it is not; the earth could be a flat slab and you’d still need the same number of satellites.