Instead of me telling you about something, I'll let someone else do it. Just because "you wonder why" doesn't mean its not there. This is from W8JI dot com.
J-pole Antenna
Because the J-pole and Zepp are electrically identical in function, and are similar to all other end-fed antennas in problems, pages on J-poles, Zepps, and end-fed verticals overlap.
The J-pole and other end-fed Hertz antennas are prime examples of antenna that
can have severe feed line common mode current problems. The coax shield has to be at zero volts potential and have exactly equal and opposite currents to those flowing into and out of the center conductor at the load and source, otherwise the feed line radiates.
When we allow the feed line shield to be part of the radiating system, due to poor feed system design or construction, the system can be unstable. With improper feed line and mast decoupling, feed line and mast length and grounding can affect SWR. Weather changes can affect feed line moisture between the outer jacket and the support for the feed line, and this can change SWR with rain or snow. Even if SWR does not change, pattern can change significantly. For example, just reversing the shield and center on a J-pole feed point can change low angle field strength several dB, without affecting SWR!
Potentially severe common-mode feed line problems of end-fed 1/2 wave antennas vary with feed line length and feed line routing. This is why some people swear by end-fed antennas, while other people swear at end-fed antennas.
The J-pole is a good example of a poorly implemented feed system, because it mixes balanced and unbalanced systems. In the J-pole, an unbalanced end-fed half wave radiator is fed by a balanced 1/4 wave stub. The balanced stub is fed by unbalanced coaxial cable. This creates two improperly treated balanced-to-unbalanced junctions. Additionally, a metal support is often connected to the J-pole antenna, adding a third variable."
I didn't know they were "notorious" for having feed line radiation. I wonder why, I've never had that problem.
If that 'J-pole' is fed as they normally are (tapped) then it would certainly require retuning, same for any type of antenna really, not just a J-pole. If it isn't 'tapped' then you would certainly have to re-tune. Simply adding torroids isn't really fixing anything, just hiding the problem.