In traditional Karate (Okinawan) tournaments were never a part of the art. As it moved to Japan, Korea and elswhere, shiai or trounaments were added. While there are so called "full contact" tournaments, most require a groin cup.
In Goju Ryu karate as taught in Okinawa by Miyagi Chonjun Sensei, students were taught how to withdraw their testicles into their lower abdomen, thus preventing injury in a physical confrontation.
The most comfortable and one of the safest ways to wear a protective cup in training or at tournaments is to first put on a standard jock strap. This will properly support and contain the nake genatalie. Then put the cup unit on over that. Most quality cups today have a softer edge region to prevent injuring the inner leg upon contact. It will also curve to the rear so that the cup cannot be driven upward.
If you are going to train for tournaments, then you should always dress as if you were going to compete in tournaments. If a groin cup is mandatory for grading, then you should always wear one during training. The more often you wear it the more comfortable it will be. I have trained for more than 40 years and the only time I do not wear a groin cup is when I am training in the dojo alone or I am training privately on a beach or without a partner.
If you are going to play the tournament game, then play by the rules. Mouth guards and groin cups should be mandatory. Other equipment should be left to the organizing body for the tournament.