The radome solves a problem of basic physics. A satellite dish is shaped dangerously like a sail on a ship. In a stationary position in small scale installs we usually solve this with guy wires. But a radome solve a more specific problem: the dish that rotates. But this creates a new problem: putting an obstruction between the dish and the signal you intend to receive. However, the problem is not the radome itself, the insertion loss from that diaelectic material is very small. The problem is the structure that supports it which can scatter signal. Robert Collin's book Field Theory of Guided Waves states that there are many applications where moderately lossy dielectrics must be used, ex. the radome. More here.
"Dielectric loss may be due to a finite conductivity of the material or to the fact that the applied frequency is too near a resonant frequency of the molecules that compose the dielectric material, so that the damping forces are such that the molecular polarization lags behind the applied field."Moorestown Giant Golf Ball." It's name was somewhat less prestigious, but it's construction began in 1960. But it was not the first either. The Bell Telephone DEW Line Radar Antenna Radome' in Alaska was built by 1958, and air mounted radomes were aboard blimps and other air craft as early as 1957. More here.
Most of this construction was by Raytheon, Lockheed, AT&T and RCA. But the geodesic dome itself beings (mostly) with R. Buckminster Fuller's patent in 1947 (granted in 1951). He built one for the USAF in 1949 and the marines in 1954. Those were simpler models, but today the same basic designs are implemented with newer materials, nylon, dacron, kevlar, fiberglass, polycarbonate, polypropylene, and various fluoropolymers.