In western pleasure, we use the term "packing the bit", where the horse is responsible for holding the bit and being sensitive to pressure.
When the bridle holds the bit up too high, it puts a little bit of constant pressure on their mouth. And it's hard to make a horse sensitive to tiny increments of pressure when their bit is constantly causing a little bit of pressure. Plus, by allowing the bridle to hold the bit for them, the horse has no responsibility to listen to the bit. Usually you will see horses "chewing", messing with their tongues, evading the bit, and just not soft to it.
When a horse is responsible for holding his own bit, a lot of the "mouthy" problems go away. The horse can now directly feel smaller amounts of pressure, and trusts that he can be responsible for the bit without pain. I have never needed the use of a noseband this way.
If by a thinner bit you mean a thin twisted wire, yes, that bit is generally designed to get a horse off of the bit. If you want your horse to pack a bit, you usually would pick one that does not discourage contact. An average snaffle with sweet iron is easy to hold, and the nutcracker effect can be avoided if you work the sides of the mouth, not a direct pull. Then they go onto things like curb bits. I like a good heavy bit and heavy reins.