All very good points, and by and large correct, though I think comparing the TDF to anything a soldier could do is a bit more of a stretch than you think, just my opinion.
But the one advantage that horse has over bike is load. Horses simply carry more. And you can always hook a couple of horses to a wagon for bulky supplies - rickshaws don't count the same - that would demand the addition of trucks to a bike unit. That said, I can very easily see the presence of large numbers of bikes in units for those reasons as mentioned in the post, but they wouldn't be in the cavalry role that horse units would.