And everyone I talked with said that it's quite common for that flap to come off (the linen threads just give way with age). I was really glad to hear that it's not all that tricky to put back on. The saddle doesn't have to be taken apart, or anything drastic like that.
It was suggested that another small strip of leather be sewn onto the flap and then that strip of leather sewn onto the saddle. It should prove to be sturdier than before be more flexible. PHEW!
And luckily, I think there are a few saddle maker's within a few hours drive that can do the repair for me. (One being the wonderful Betty-Lynn Tattersall). I'm actually thinking I could likely do some of the repair myself, veeery carefully of course!
Thanks for all of your suggestions on people to help me fix the saddle! You guys are great resources!