I want to congratulate you on being as well adjusted as you appear to be. Most of the guys here (especially the younger ones), can think of little that would be worse for them than gyne. I'm old now and can assure you that there are things in life that are FAR worse than having gyne, although I wouldn't have thought so when I was young. At that time the thought of wearing a bra would have been incomprehensible.
I am one who wears a bra every day to work for the support and for the comfort. My gyne came on more recently due to hormonal imbalances and testosterone injections, so I didn't grow up with the fear and intimidation that many here seem to experience. Perhaps that why I tend to see a bra as merely a practical solution to the problem. I am quite uncomfortable with my breasts swinging, bouncing, jiggling, rattling, or whatever as I find it distracting and always draws my attention to them. I have found that a bra provides the support i need and allows me to go through most of the day having completely forgotten that i have breasts at all. Another thing that a bra does is make them less "pointy" when my nipples are hard, and therefore make my breasts less noticeable.
Congratulations on being mature enough to understand that you are not your body, and that it is ones character that makes the man. If you have found that wearing a bra makes dealing with your gyne more tolerable, then by all means wear one. I do, as do many others here.
Anon