In my opinion the answer is that they should not, unless they have a history of back injury. The simplest reason for this is that they do not compete in their sport with a lifting belt (except for Olympic Weightlifting, powerlifting, strongman and Crossfit). As well the belt is often used as a crutch by many in the weight room. They rely on the belt as opposed to utilizing and bracing their core musculature. This in turn will lead to a reduction in recruitment of the core and an emphasis on chasing numbers. I'll use the 2016 crossfit games deadlift ladder event as an example, particularly the women where they all wore belts. Many had zero regard for keeping their form on the lift looking more like the Gateway Arch in St.Louis than a deadlift. This put their spine in an extremely compromised position (see video here https://youtu.be/SuSS7YD8SAw). Interestingly the woman who won that event maintained excellent form throughout all lifts. Coincidence? I don't think so!
What does this mean for athletic performance? The core musculature's responsibility is force transfer from the lower body to upper body and vice versa. If the core doesn't develop strength at the same rate as either the upper or lower body, the body cannot realize it's actual potential. That's because the transfer point is the weak link in the metaphorical chain. This doesn't matter what movement it's a sprint, throwing a ball, or a punch/kick because the body will not use its potential for force output.
The argument for using a belt typically is that of "it's safer for the back" isn't necessarily true. It's never safer to lift more than a weak link in the chain can support. This will create strength imbalances which act as a ticking time bomb for injuries. Lifting without supportive equipment causes all parts to recruit and improve. Minimizes the potential for an imbalance, especially when taking an exercise through its full range of motion.
But what if there is a lagging part or sticking point? This is where isolation, partial range, pauses, and other methods are used; to strengthen the weak part. There are times when a smaller muscle is the limiting factor in progress in a lift. This is when direct isolation work is required.
So in my opinion athletes should park the belts on the shelf. Focus on perfecting technique and learning to feel the movement. That will yield greater results and transfer to their performance.