If the antenna system was designed to use the coax shield as a radiating element, then yes ferrite on the coax would rob some power. An end fed antenna using a non resonant length of wire and a 9:1 balun would be an example of an antenna that uses the coax shield as a counterpoise, but you have to choke those near the radio to avoid hot RF on the radio and mic.
A vertical antenna with a single counterpoise wire is actually a dipole and that single counterpoise radiates, and choking off the coax to prevent RF on the coax and coax radiation would be a primary goal and would not rob any useful signal that would radiate as a stronger signal. For a vertical antenna with multiple counterpoise wires or a ground plane, the ground plane or counterpoise does not radiate. In this case a primary goal would be to insure the coax does not have RF flowing on the shield and adding a choke balun would not rob any useful signal that would radiate as a stronger signal.
The counterpoise in both above examples connects to the ground side of the antenna feedpoint and a 1:1 ferrite choke balun would have no effect on RF currents flowing into a counterpoise since the choke balun is downstream from the feedpoint.
In your example of the vertical over poor soil, I'm assuming its a vertical attached to a ground stake with no ground wires or counterpoise and that leaves the coax shield as the only counterpoise. That is a situation you don't want to create and if so choking off the coax will degrade performance because someone made a mistake installing a vertical with no counterpoise, leaving the coax as half of a dipole laying on the ground and radiating all the way back to the radio. I spend at least several days a week playing with choke baluns and fixing other peoples RF problems, so its a very familiar topic for me.
The firestick is simply a short loaded antenna and they do have an optional gimmick coax feed for mounting to truck mirrors with poor ground plane. The no ground mount came out after I left the CB market around 1980 and I have virtually no experience with it. If the OP is using the no ground cable, then that may explain the changes when he inserts a meter and jumper since that cable harness is probably a tuned length. In that case I might try a regular coax run that grounds to the tool box.
Otherwise I've installed hundreds, possibly 500 firesticks and compared them side by side to everything on the market at the time and they work about the same as a similar length Wilson or base loaded A/S or Turner was popular back then, or Motorola, or you name it. I swapped lots of antennas doing dozens of range tests between every antenna available and for its size the Firestick holds its own against anything of the same length.
My first choice would be a Larson or similar roof mounted, but the OP indicated he can't drill a hole, so a Larson is out of the question anyway.
The use of ferrites to absorb (convert to heat) the errant current flowing down the coax does indeed rob useful signal that could otherwise be put into a proper counterpoise and radiated as a stronger signal.
It is like having an HF vertical on poor soil when you sop up signal into ferrite. It will tune up, but the signal will be weak.
The Fire stick antenna is a gimmick antenna unless it has an effective groundplane. It is marketed with mounting and feed options that are inferior to a properly installed NMO27. I don't think there is any cost savings whatsoever.
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