My answer is a clear YES! Though like in everything in life you should never say "never". There can always come up a case where it is ok or even better to skip a noseband, at least for a certain amount of time. Generally, I ride always with a noseband because I believe that it helps the horse to give the "right" and easiest answer to my aids.
What I mean by that I will try to explain in this example:
Take one of your hands that and compare the area between your thumb and your index finger to the mouth of the horse. Of course, there is now tongue but the joint between the thumb and index finger is quite similar to the corner of the mouth of the horse. First, you put the thumb and the index finger together like the horse would have a softly closed mouth. With the index finger of the other hand go and pull, like the bit would do in the corner of the "mouth" (Pic 1 - see below) How does that feel, even if you pull a bit harder? Now pull the thumb and the index finger apart, like the horse would open the mouth widely. Pull again (Pic 2) and compare this feeling to the first version. For me, the second version hurts while the first is quite ok. The horse has a reflex of opening the mouth when the pressure does occur in the corner of the mouth. (That reflex we use forex. when we put a bridle in his mouth). But without a noseband, many horses tend to open their mouth too much by reflex and that will hurt them, like the second example of your hand showed. Especially when we start to train a young horse, they don't know what we want from them, so they give us many trial error reactions. Good trainers know how to put up the aids so that it is easy for the horse to guess the right answer in the first place. By repetition, the horses learn and we must take care, what repetitions they are "training". So by coincidence, it is easy to "teach" the horse to open the mouth widely and even worse than it learns that the bit means pain (even light rein aids become painful with an open mouth.
Of course, there might be horses that are not reacting by opening the mouth widely (trial error). But in my opinion, it is better to be safe than sorry. If there is a noseband and the horse tends to open the mouth too much the noseband will stop this behavior and it needs to find another answer, which is hopefully soft chewing, slowing down, and getting soft in the top line.
An old horseman in Iceland, that grew up during times, where the horse was very important for transportation told me once his maybe not very scientifically proofed conclusion but an interesting observation: He observed that the horse tend to have the mouth slightly open while moving and it is more relaxing on longer distances for the temporomandibular join if the lower jaw can rest on the noseband. Since it was quite common to ride with an Icelandic kandarr more than on a snuffle bit it was also more common to ride without a noseband. For some reason, he preferred to ride with a snuffle bit but when he used a softly closed noseband the horses got less tired in the neck in his opinion. - Sounds somehow logical to me but it would be interesting to do research this further.
Like it said: No tool is as dangerous as the hand that is using it.
Of course, a noseband should never be used to close the jaws of the horse too tight together and restrict soft chewing and swallowing. Like everything in life, we need to find the middle way according to each individual. During my BS study in horse training and riding teaching, I wanted to know more about this issue and wrote my Bachelor thesis about: "A comparison of different methods of estimating noseband adjustment". (see below). For sure a lot had happened since then, but I hope you can find some interesting points in it.
I think in general it would be good to have a certain tool for this measurement in competitions since there it could cause a disqualification if it is not properly done.
In daily life I handle nosebands and their adjustments after these rules:
If you want to read an interesting article about the history of nosebands checkthis article
Best greetings for today. Hlín