I agree with Anabel a little bit, but I'm a bit of a fence sitter. While rider imbalance probably isn't the only problem here, it probably a factor.
I consider myself a very good rider, I've been riding for eight years.. I've ridden very unruly horses, horses who've reared, bucked, fell underneath me, kicked at my horse, I've been bucked off and trust me it takes a lot to get me off (One of the first few times I mounted my filly alone [I had done it on a lead the first couple times and she was fine] I had one foot in the stirrup and wasn't even fully swung over her back yet [keep in mind I sit lightly too] when she went bucking bronco style on me and I probably hung on for a couple bucks too many. Laid with the wind knocked out of me and a huge bruise Over my Kidney. I got up grabbed her, mounted again (praying she wouldn't pull the same maneuver, but I had a good/better grip on the reins and rode her for about as long as my kidney would let me. I'm always given the most difficult, stubborn, slow, lazy, spooky, cookoo school horses and for the most part can conquer them in a few lessons. I've ridden horses everywhere from green to finished. I'm not being cocky, but I consider myself an accomplished rider. I agree with Anabel, riding is about humility. I thought I was a balanced, good rider. Even bareback..
Then I was stripped of my stirrups and reins, and forced to put one ankle up while I held both arms out (Posting, I was on a lunge line and a very safe school horse). Then I realized what my trainer was talking about, I leaned into my left shoulder. Try that exercise in a lesson on a safe, mostly balanced horse. Switch legs, go both ways.. put one arm on your hip and one arm out, switch. If you can do that for about half an hour without losing your balance once, then yes you are a balanced rider. If not, then you need improvement: EVERYONE DOES! I probably grabbed the pommel of the saddle every 3 seconds. Then the gap got larger and larger, until It was several minutes. I gained more and more balance.
I'm not trying to insult you, but you don't ride perfectly: No one does. Humility is trying to get better every step of the way. While your balance may not be the whole reason this horse is imbalanced, I'm sure it plays a role. A good rider recognizes the limitations of their horse, and tries to strengthen their own weaknesses in order to make-up the slack for the horse.
The more you work on your own balance, the more you'll help him.. If he isn't conditioned and needs to gain muscle, you have to interfere with his balance as little as possible. Lunging and flat work will help a lot, but they gain the most muscle by actually having weight on them. I lunged and long-lined my filly for five months religiously before riding (training) her, granted she's only 3 and I haven't been riding her since I've been at school, but even before I left she needed to build a lot of muscle. I try to interfere with her as little as possible, and just let her get used to my weight and build muscle. I ask her to collect a little bit and just do her job, which is to go around at certain speeds, make circles and serpentines, halts, backs, and turns on the forehand and haunches. If I push her before she builds muscle, she's just going to fall apart and build muscle favoring one side, or be really weak and imbalanced in general.
My suggestion is riding him as much as you can in a week, giving him a day or two off at least and keeping some days a little light (like half of your normal routine) to give developing muscle a chance to heal. Have two days in a row where you work him hard, give him a day off, do half the work the fourth day, and so on. Let him build muscle, keep him bent around and on the bit, forcing him to carry himself (not rundown his front end) and lift up behind. You'll see how much balance they magically gain just by keeping them collected and traveling underneath themselves. I'm sure he's still imbalanced after that, so build his muscle tone and work on your balance.
I once rode a horse who had ringbone (but was sound) on one or both of his front legs. Everyone who rode him couldn't get him to move and found him to be extremely imbalanced at the canter. (It was at a therepeutic riding center so a lot of the kids were volunteers and had limited experience) I rode him and got him working behind, on the bit, and moving! He was balanced at the canter and didn't fall in (most of the time, the canter's the toughest gait for me to help with imbalance because it's where my lean is the worst). I'm just saying, the world of difference a little bending and flexing could do. (I counterbent him for a little, then did circles out of each corner, at the trot, after that he started to loosen up and bend around)
In order to find out how to improve something, you need to first understand why it needs improving. If your part of the problem, then you need to improve before it can get better. I'm banking that working on you and the horse simultaneously will dramatically improve his balance. Whereas if you just work solely on the horse, you will just build him up to the point where he's making up for where you lack.. So he's covering up the fact that your not necessarily balanced.
I'm sorry, but 6 years is just icing on the cake compared to the best trainers' experience. The horse industry is a tough one in that, every horse you encounter teaches you a new thing! No matter how long you've been experienced in the horse industry, you'll always encounter a new problem.. or an old problem where the traditional technique doesn't do the job.
So in Anabel's defense, I urge you to take a second look at yourself. There's nothing wrong with admitting your flaws, and being unbalanced as a rider is a very common one, and pretty simple one to fix (simple as in the solution is there, but it will take work and time to do). Even some of the older, more experienced trainers and horse lovers here admit their weaknesses to each other.