There are a bunch of different things you can do on the ground.
There is in-hand work, with either a bit or a halter. This is structured work where you are teaching flexions, lateral work, turning on haunches and forehands, half pass, backing up, etc. in a disciplined way that will lead to doing the movements in the same way under saddle. I can’t imagine how you would teach any of this from the saddle, without doing it quite extensively on the ground. It is also a time to get the horse used to the bit. I associate this method with dressage, though western trainers also do it. Here, you are asking for precision and correctness of movement, and the horse is not given a great deal of freedom during these exercises. It is like schooling the movements on the ground.
Then there is “ground work,” in the sense of obstacles, bomb-proofing, ground-tying, and good leading skills. This is more connected to western training, though all horses benefit from it. You’d do this in a rope halter, most likely. Here the horse is given more freedom how to perform the movements, and is expected to behave. By freedom, I mean that if you are teaching a horse to walk quietly over a tarp, you have them on a loose lead rope, and you are obviously not simultaneously trying to micromanage the quality of their walk.
Then there is liberty work. When trainers who specialize in what I’m calling “ground work” go to horse expos and put on a performance, they often show off liberty work, despite the fact that this is not what they actually teach in their clinics. This is also what the Cavallia troupe do: horses running large, running patterns, coming and going to the trainer, while stirring music plays in the PA system. It looks amazing, and people always say “I wish I could do that with my horse.” But real liberty work is also highly trained work, even though it is done on the ground. It is qualitatively different from “in-hand” or “ground work.” I would say that if you are reaching a problem spot with liberty work, that you reach out to a local trainer who specializes in this for some lessons. I am guessing there are holes in your earlier training that are appearing now as you reach some speed.
What are your goals? Are you just doing liberty work to have something fun to do with a young horse? In that case, back off the parts that are becoming dangerous and return to ground work, which is more directed.
Or do you have a time line to have a finished liberty horse, for performance or to do things like agility classes? In that case, get some professional help.
As far as whether work on the ground helps work in the saddle, I would say absolutely that dressage in-hand and western ground work both help enormously. I don’t however think there is much direct benefit from liberty work.
For my current program, the dressage in-hand work is the one that ladders into my riding, and that I’m responsible to my coach/trainer to do correctly and systematically.
I’ve done the ground work obstacles and bomb-proofing, maresy is good at it, but she’s a confident horse, so we don’t need to school this forever. I did a good ground work clinic early on, learned a lot, and incorporate the common-sense instructions into all my interactions with horses.
I also do clicker trick training, which is very effective on the ground with this horse, but rapidly gets counter productive in the saddle.
I’ve done bits of liberty work just for fun in turnout, watched others play with it too, and my guess is that not all horses are going to have the talent for it. You need a horse with a lot of energy so it is willing to run off on cue, but also with a “draw” to people, so you can call it in. And you need a horse that is not going to get aggressive up close. Cavallia picks its liberty horses as carefully as any performance rider picks their prospects, for sure.
I’ve got a big, heavy, dominant-personality stock horse mare, and I’m not comfortable up close while she is getting out the wiggles in turnout. I certainly don’t want her running full blast at me and then doing a sliding stop. In fact, I want her to always approach me at a walk in turnout. I stand back and let her do the full buck n run n squeal n fart thing for ten minutes or so, let her calm down on her own, before I get up close and ask for moves or tricks at liberty. That works well, but it means that I’m never trying to direct her full energy at liberty, which means I’m never in a position to try the real send and draw aspects of liberty work, only slower, quieter, things. So I don’t think I’ll ever be doing the Cavallia style high-energy work, which is fine.
On the other hand, my friend who has a lighter, handier, Arabian gelding can get him cantering around her on a 15 metre circle, following her at a collected canter while she jogs a circle, etc. He just has the energy, the physical ability, and the personality to do this almost naturally.
OP, just don’t try to do it with spurs on.