I agree. Now, I have 0 problem with bitting up for shows and special occasions (like hunting), but IME most horses that are rushy/heavy at home are either unfit or lacking basic flatwork. Usually both. Even the worst conformed animals should be able to pop around a little course calmly and without pulling.
A lot can be accomplished on the flat, but there is something to be said for low, technical gymnastics. I wish I could find the article but I believe it was Beezie who would give Big Eq and other high level junior riders clinics at 2’3" and absolutely kick their backsides. If you can’t do it over speed bumps, there is more work to be done before putting the fences up. This would also give OP time to work on their hands, doing no reins and no stirrup trips through grids and working on the flat.
ETA to the bit question specifically - I do like a Waterford for horses like this when needed, or maybe a Myler with hooks. With really good hands come more options for specific instances, but sometimes just switching to a double joint or really soft bit actually does the trick while the rider reinstalls the buttons - bigger isn’t always better.