I haven't read the article and, though vegan for a long time, have never heard this issue mentioned before. While I can't speak for all vegans, I can tell you what I think.
You are certainly right that the ways in which these animals have been bred and trained is not "vegan" in that is the systematic exploitation of animals for human gain. I imagine most vegans see riding a horse as an unspoken endorsement of that process. Also, most of the time when people are riding horses, they either purchased them or are paying someone who did purchase the animals. This is no longer turning a blind eye to the industry, but actively paying for it.
I don't think most vegans have your hypothetical of rescuing a horse from an animal sacntuary in mind when thinking of riding, as it is a very small percentage of the people riding horses. In the unlikely case that a vegan happens to now own a formerly abused horse and would like to give it exercise, they might ride it, or they might find other ways of exercising, still seeing riding the animal as using it, even if the animal has been bred not to feel that way. This is such an isolated case that it's hard to say what vegans would do.
It should be mentioned that the most widely accepted idea of veganism includes not only not consuming animal products, but also the notion of not using, exploiting, or imprisoning animals for our tastes or gain. With that, I think that most vegans would have a problem participating in any sort of horse industry, but that is only my opinion.