There are many subjects, including engineering, architecture, agriculture, medicine, cooking, and music, that were the result of thousands of years of trial and error, figuring out what worked best, and then just carried on through tradition and cultural practices, passed on by word of mouth, or the older people showing the younger people how to do it, without knowing how or why their practices, methods, or tools worked the best. Then, at some point, people start thinking of it differently, employing a bottom up approach, with increased scientific understanding of the underlying mechanism, with new scientific understanding of physics, chemistry, or biology, calculate from first principles, what would be the best way to do it, based on your understanding of how and why it works. This is always a major turning point because it is at that point that humans can actually take control of the subject, and design it to get a predicted preferred outcome, because for the first time, they actually understand what they are doing.
The history of sports followed the same pattern. All sports evolved over the course of hundreds of years of trial and error. People would modify existing sports by varying the parameters, such the number of players, the size of the playing area, the duration of play, the size and weight of the ball, modifying the rules, and they would find that the modification would either make the game less enjoyable, in which case, it would be discarded, or more enjoyable, in which case, it would be retained. The details of the rackets in the various racket games you listed is the result of centuries of tedious trial and error. You ask "What would happen if we used the wrong racket for a sport?" Answer: "It would not be as good. That is why we don't do it".
We are lucky that sports equipment design has finally made the transition to engineering design calculated from first principles, based on the underlying physics. APS News had an article discussing the physics calculations that went into the recent redesign of the official soccer ball, called the "Telstar 18" used in the World Cup. They did a lot of physics calculations, computer simulations, and experimental testing before they settled on the final design. You can read it here.
Bend it Like Bernoulli
https://www.aps.org/publications/apsnews/201807/bernoulli.cfm