I don't think there's a problem with your tv. Though you said you tried adjusting everything, I have to ask: have you tried the frame smoothing option? It should smooth out flickering from low frame rate video with fast movement. If yes, the next step is disabling any picture sharpening enhancement if it's on.
The last thing to do (in order to attain best possible image without flicker):
If you haven't tried lowering the Sharpness setting from the center, default, position, do so until the flickering subsides enough while still leaving enough detail. It shouldn't require setting it to 0, maybe 25-50% the way there.
Make the adjustments in an unused picture mode you reserve for watching the game, whether it's sports mode or a cinema mode - the one you get the best results with, without using the best one for movies/tv obviously
Edit: I have a Sony TV and the sport mode degrades colour accuracy compared to other modes. I don't watch sports, but instead of using the sport mode as an alternate picture mode for certain content, I use the Animation mode (not for animation) because it's almost exactly the same as the cinema pro mode. You might want to do the same with one of the options of the CX.
If you can't get the problem fixed after everything you try, and you're not a gamer needing 4k 120fps compatibility, Sony's image processing and motion handling of video is notably better than LG's, so you could get an A9G (has 48" variant I think). Next year's model will probably do 4k120 if you can wait and need it.
It may be your source too. Unlikely to be a cable/satellite box, but if it's a cheaper thing like a firestick or a built in tv app running on a slow SOC it could be. AppleTV seems to put out consistent frames - if that or a good blu ray player is the source and you still get flicker it's definitely the tv
