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Check out Peckham BMX club. Kye Whyte representing tonight. Club does great work to get local kids into the sport.
It's tricky though as to be competing in the BMX national series here now requires a pretty hefty investment in kit, entry fees & travel + accommodation through the summer. Not quite as exclusive as dressage, but perhaps no longer quite as inclusive as it was a decade or so ago.
When it comes down to it though, pretty much anything outside the OG Olympic sports and martial arts requires money. I get that Dressage may be peak, but my bet is that the stories of people from poor backgrounds are the outliers.
My only knowledge of dressage comes from an ex who had a horse and stable shared with a mate to keep costs down. She had a decent job, but wasn't minted or from a wealthy family. But then by the time she had the money to keep a horse, she'd probably have been too old to develop the skills to compete...
...so yeah maybe you're right.
But ultimately all professional sports are about exclusivity not inclusively.