The exhalation is not linked to any particular sports, but a rather profane mechanism: Tensioning of the abdominal muscles reduces the volume of your lower abdomen and thus pushes your diaphragm upwards, which in turn reduces the volume of your lungs so that the air has to go somewhere, voluntarily or not. Any kind of lifting, crunching, or torsion movement will involve the muscle belt between hips and ribs which is the actor in said mechanism. So this definitely holds for throws as well.
If you have breathed in and close your windpipe consciously when making these movements - especially against resistance as opposed to fluid movements - you will notice a strong pressure in your throat and abdomen because of compressed air excerting pressure from within your lungs, effectively resisting the compression from the outside. This pressure can become high enough to make your alveoli pop and hurt your lungs. Before that happens, your body automatically loosens muscle tension to protect your lungs from injury. This involuntary relaxation, in turn, will inadvertently lessen the power of your move.
This is why we train relaxation and breath flow in martial arts: To train not to have so much tension that we close our windpipe, ie. "hold our breath", when we actually should not. Breathing in and/or stopping the free flow of air is a natural mechanism for "bracing" for some kind of force, but it is detrimental to the outcome in all martial arts situations, may it be striking, throwing, falling, or being hit. A secondary effect is that if you consciously breathe out forcefully, you engage exactly the muscles you need for the proper transfer of force through a stiffened frame.
Therefore, regardless of the level of the practitioner or the sport, if you do not exhale (or have exhaled) in powerful movements, you make your body work against itself, no matter how you describe the mechanism (ie. in terms of physiology and physics, or qi, or whatever). The reason why doing a pronounced exhalation at this moment exactly is optimal is that you support the tensioning needed and lose less time with air inhaled, ie. you can generate more energy over time. Especially the latter is a very important factor in fights, which extend over time, as opposed to one-off movements.
In other words: The more power you generate through your abdomen and the more powerful and longer the engagement is, the more important it is to exhale consciously. In any case, you will want the air in your lungs to be able to get out when you need tension in your abdomen. Since powerful striking and throwing which does not rely on fluent full-body movement but acceleration from a stance instead generates a lot of its power through the abdomen, you will want to exhale doing it. This will not be similarly pronounced when generating force through bigger and/or more fluent movements, but for the moment of contact or final acceleration, it should and - if executed properly and with great power - still will be there.