It’s tricky because the horse as well as the rider is the athlete, whereas in other sports it is human athlete plus gear. It is easier to ban a certain type of equipment across the board than it is to ban a certain type of athlete. No one wants to put a height limit of 5 foot 10 inches (modern average height for men) on basketball players!
There is no “level playing field” in respect to the aspects of a sport that rely on physical ability, and in riding the physical ability of the horse matters, in the end, almost more than the human’s ability (proof being that an international level rider might not make the Olympic team in a year when their top horses have all gone lame).
There is no level playing field in any horse sport, if you mean that the lower level horse can go in and realistically compete against the better horse. There are efforts to do so, for instance TB racing tends to divide up in tiers based on the quality of the track, and then have multiple classes with all kinds of restrictions. Hunters have a series of green classes. Having show classes split between AA and Open is supposed to accomplish this, but of course as everyone points out, there are deep pocketed AAs out there who buy a high end horse, have it completely under the care of a BNT, and just get on to pilot it, and would probably beat most smaller trainers who are riding project horses.
So really the quality of the horse is the single biggest factor in whether you are going to get ribbons in any horse sport. Its true that the giant gaits of the top WBs skew dressage towards effectively being a breed show, but even if you stopped weighting giant gaits so strongly in the marks, there would criteria. If there was more emphasis placed on collection, that would make riders of Andalusian horses happy, and might put some QH in the arena, but it would still favor horses with natural collection.
So I don’t really have an answer.
OK, I see now that OP wasn’t really thinking levelling the playing field was possible
so maybe all this is off topic.