I don't want to sound mean, but without any links to anything, any mentions of books, papers, even mentions of archaeological finds in local newspapers... what you're saying is just a story about someone you met who made some outlandish claims.
The fact that the argument appears to be cut and dry is also testament to the fact that it has been rehearsed many times and shared within that identity community
The kind of horse that could be domesticated and ridden was not widespread, and in fact were really just hunted for meat until maybe 6500 years ago, and actually riding them came even later...
So, I mean, maybe there were equine species in North America, but their relationship to humans I doubt would be anything like what we think of horses like right now. More like a hunted species like bison or deer...
Now, dromedaries I can see. Camels, alpacas, and llamas etc. Long and ancient history with those, and in both hemispheres independently.
Edit: more like millions of years
Both caricatures ofc. But then again, they have to be because in the political arena details and facts don't matter at the end of the day as it's all about marketing.