What I find interesting about the double bridle is how opinions on it have rapidly changed. It wasn't so long ago in England that snaffles were considered fit only for grooms, children, young horses and the mutton-fisted! Everyone else would only be seen riding in the double, especially on the hunting field.
I personally do not ride in a double bridle but I do ride my pony in a hartwell pelham with two (or four, depending on how you think of it) reins. I ride him in this in the school and out hacking as well as over the odd jump. He also has a sharer who rides him in a snaffle with a flash noseband and martingale. She is a child and would find 4 reins difficult to cope with and he is can be quite hot headed. In getting my pony used to his pelham, I followed the advice of Laura Bechtolsheimer in relation to double bridles which was to start off by hacking your horse out in one before moving onto simple schooling. I don't really understand the idea of only putting on a double bridle for competition or for the odd training session. If the horse is going to be comfortable and you as a rider are going to be able to use it correctly, you need to be training in it regularly. I also spent a few months practising with four reins off the horse so that I would be able to handle them once riding and had lessons with a good instructor to make sure I was using it correctly.
The way I use my pelham with two reins is essentially riding off the snaffle and allowing the curb rein to be activated by my pony: e.g. If he is going correctly, he only feels the weight of the rein with no contact there. If he pushes his nose too far in front of the vertical, he brings the curb chain into effect by himself. This way he essentially teaches himself. I use this bridle for showing as I do not compete in dressage.
My definition of a schoolmaster (as it was discussed earlier and the definitions given were different from mine) is a horse or pony who is well schooled and understands the aids for his game up to the level he is trained to. He is not necessarily quiet (my pony, for example, was a JC showjumper and FEI event pony but is not quiet to ride and can be very hot headed) but if he or she is given the correct aids and ridden in the correct manner, he or she will respond correctly. This is why true schoolmasters are often not novice rides. A true schoolmaster is different to a quiet or beginner riding school horse, for example, and has often competed at a decent level with success in his or her discipline. They are usually used to teach competent riders the more specialist aspects of a sport e.g. How self carriage should feel or how to execute flying changes correctly.
So, essentially, a true dressage or showing schoolmaster is ideal to use for teaching a rider how to use a double rider in the presence of a good instructor who understands the bridle and the bits. This is because they are already used to the bridle so you can concentrate on yourself and your riding. If you want to use a double or even a pelham with 2 reins, this is what I would recommend before trying to train a horse who isn't used to it in one.
Also, reading about how the bits work and watching videos on this (including different rein positions) is actually invaluable before you start. I know some people think only practise helps, but I disagree. Knowing how these bits operate mechanically and what you are trying to achieve is absolutely essential in my opinion. I see a lot of people riding in doubles or draw reins who do not know what they are doing and their horses are going round with their noses in their chests rather than in self carriage and it is very sad.
But, to answer the question, there is no reason why you cannot ride in a double bridle every day if it is used correctly and your horse is schooled to understand impulsion, straightness, rhythm, direct and progressive transitions and is responsive to the leg, body and hand in the snaffle. I also believe it can be useful as a training tool in so far as you school a horse thoroughly to novice/elementary level (UK levels) before introducing it as you can continue your training from there. Of course, you do not rely on the bridle or expect it to train your horse but it can then help you with your flying changes and collection used correctly as you can use it to refine your aids and give clearer requests.