First of all, I would not ride any older horse is a halter unless it was impeccably trained and well-mannered and then I would not ride it often that way. It is impossible to school a green or a spoiled horse in a halter. I have seen people try, but they were usually just being drug around where the horse wanted to go while the horse just learned to argue and win.
If it seems to know what it should do on the ground, I saddle it, put a full-cheek, 3-piece snaffle on it and let it stand around all day for a day or two.
A person has a lot more leverage if they need it and a lot more control when their feet are on the ground rather than just being a passenger.
When you ride an uncooperative horse in equipment that does not give you any advantage, all you are doing is letting them practice being disobedient and teaching them to argue. These are very counter-productive to ever having them be worth while to ride. A poorly trained horse has no more value than an untrained one -- or even less.
An "interesting" point of view.
Having studied with James Roberts, a professional colt starter and re-starter for difficult horses, he generally did not advocate riding in a bit until the horse has around 100 to 150 hours riding time.
My young tb mare that he started for me is a case in point. I only around 15 hours on her and have yet to use a bridle. I just use a rope halter and rope reins. In fact to get her thinking forwards we even stopped using the 12 foot lead rope we started using, the heavy clip was too much interference around her face. She goes much better in just tied rope reins.
She will be ridden in a snaffle bit, but only once the schooling is to the level that the bit is used very the subtlest refinement, not for gross control.
What sort of advantage is it that a bit gives you ? Having re-started riding after a long break in just the last few years I have come back to it with fewer "received" truths and find myself questioning more things. The use of the bit for control of an unruly horse is one.
I watched a very good rider and friend of mine helping a young girl school her horse for jumping. It kept ducking out to the side. Eventually Liz decided to have a quick ride and removed the bridle to replace it with a halter. She reckoned she would actually have more control in the halter, and so it turned out. The horse was very soon jumping beautifully.