I think keeping everybody together was the real reason. Also Frodo wouldn't leave his friends in danger and only mounted when Glorfindel convinced him that they were less in danger if they weren't with him.
I believe the BR's would have followed Frodo and the ring and left the rest of them alone. But still...I was forgetting that they would be needed to deal with the Riders left on the wrong side of the Ford. [what do you need with a Ford when you've got a horse? Hehehehe]