Ground and Earth are rarely the same thing. All ground usually means is *circuit common*. While the Earth can sometimes serve that purpose, it is a big old lossy sheet resistance, and often isn't a very suitable circuit common.
An AM broadcast antenna is one of the few examples where Earth serves as the counterpoise for the antenna. But Earth by itself is very lossy, so copper
*radial* wires are buried in the Earth, radiating out from the base of the tower like the spokes of a wheel. The transmission line is connected between the tower base (insulated from Earth) and this system of radials. 120 radials are usually used. Sommerfeld figured out in 1932 that a system of 120 radials, each 1/4-wave in length, would reduce Earth loss to a level sufficiently close to a solid copper Earth as to be insignificant in engineering terms.Rod type car antennas work similarly, with the car body substituting for the radial system. Earth doesn't matter here. The car could be suspended thousands of feet (or miles) above the Earth, and the antenna system would work just the same.
The old electrical maxim is, "It takes *two* wires to make a light". Cliche as it sounds, it applies to antenna systems too. A coaxial transmission line is a two wire line. Current is driven in one direction on one wire while it is driven in the other direction on the other wire. Charge is conserved. Symmetry is preserved. If this weren't true, you'd just have an open circuit, essentially infinite impedance, and no currents would flow.
Antennas are subject to the same rules as transmission lines *except* that they are purposely designed to be anti-symmetric. In other words, they can be seen as an opened up transmission line where the electric and magnetic fields aren't canceled by physical symmetry of the equal and opposite current flows. Thus the fields can expand out into space as electromagnetic radiation fields. (Or vice versa, ie an external field can induce current flow on the two parts of the antenna system, providing a +/- driving current source for the two wire transmission line. Antennas are reciprocal.)
Every antenna system has to have two parts, call them a poise and a counterpoise. This is most obvious in a *dipole* antenna, which has two equal arms extending in opposite directions. But the "arms" don't have to be physically identical, there are many different types of antennas with different physical parts. But every antenna must have this poise, counterpoise, duality.
In the case of most car antennas, the antenna rod is one part, the body of the vehicle is the other part. The antenna can't work properly if either part is disconnected or missing. Now the *usual* problem in cases such as you describe is that the coax has deteriorated. Either it has gotten water in it and corroded, or the inner or outer conductors have broken, or shorted to each other.
In any case, the repair is the same, replace the coax. Now the coax used in auto antenna systems is high impedance. The inner conductor is a fine wire, the outer is mainly foil, and it is a real bitch to make sound connections on each end unless you have the proper (Motorola) crimping tools for it.
Since replacement antennas, with coax already attached by the antenna manufacturer, are reasonably cheap (usually under $20), the usual repair is to simply replace the whole thing as an assembly.
Now note well that some cars use *active* antennas. These antennas have a preamp built into the mount of the antenna. The coax feeding this sort of antenna is usually low impedance (50 ohm), and has DC to power the preamp phantom fed down it. If that's what you've got, then the replacement radio has to be able to feed DC down the coax, and it must be designed to accept RF from a low impedance source. The vice is also versa, if the replacement radio expects to work with a passive antenna fed with high impedance coax, it won't work properly when hooked to an active antenna system. In other words, the radio and antenna system have to be compatible with each other to work properly.
Gary