IMHO, a well trained horse will ride in whatever the heck you want to ride them in. I can ride mine in a bit, or a halter, or nothing more than a string around their neck. Personally, I would never look at buying a horse that was advertised as "Never worn a bit" or "Trained in a halter/bitless bridle/etc" because, to me, that horse is missing a huge section of training.
Whatever you put on the horse's head should never be about "control" and if that's what you're using it for, then you're using it wrong. When a horse gets to the point of being well trained, then the headgear you choose becomes all about subtlety. A neck rope doesn't offer as much feel or subtlety as a halter and a halter doesn't offer as much as a snaffle and a snaffle doesn't have as much as a curb.
On one end, you're trying to have a complex conversation with your horse using those old play telephones that everyone made as a child...you know, with the 2 tin cans and the length of string? You go into different rooms and pull the string tight and talk into the can back and forth. Yeah, it works, but you have to speak relatively loud and things still come out muffled. The range of communication is very limited.
On the other hand, you've got a good quality digital landline. You can whisper into the phone and whoever you are talking to can hear every single syllable, regardless of whether you are 5 feet or 5000 miles apart.
I'm all for folks who wish to ride their horses in halters or sidepulls or bitless bridles or what have you, but I do get very tired of everyone just assuming that those of us who choose to use bits are automatically hurting our horses, that the bit itself causes pain regardless of how it's used...that every horse would be happier "without some hunk of metal in their mouth". Unless a person has experience with every single horse everywhere in the world, then blanket statements like that are just straight up incorrect.
My horses do not get hurt with the bit. If they did, then they would not be nearly so quick to pick it up out of my hand as if it were a treat. They would resist being bridled and they would be cautious and hyper-reactive every time I picked up my hand to give a cue (the way horses are when they have been handled roughly).
I guess I just subscribe to a different way of thinking. So many people think "I must use the absolute softest headstall on my horse so that there is no chance of ever hurting him", whereas I feel like "I should work to improve my ability and my hands so that I can ride my horses in whatever I choose with no chance of ever hurting him".
IMHO, a person can never hope to advance beyond a certain level (or advance their horses beyond a certain level) unless they are willing to challenge themselves to learn to effectively use more advanced pieces of equipment.