In that time, people didn't have all of the clothing stores that we have today. Of course, they could buy clothing in markets similar to today's flea markets. But clothing was something that was expensive or took a lot of time to make. They would not have wasted clothing. They wanted to keep the victims clothing for themselves or the soldiers families or to sell.
The bible confirms that they took and saved Jesus' clothing for themselves. Matt.27:35. Therefore the bible confirms the real reason for nude crucifixion. NOT shame.