Everyone should be able to enjoy riding their way. I'm a trail rider because I love nature. On a hot day, Dixie's favorite place is in the river in a shady spot and I think she'd stand there all day if I let her. People can come and go and she won't try to follow.
One day an island we were riding on was covered with TINY frogs - some so small you wouldn't see until they moved they were so small. We've stood in almost reaching distance of deer - if you stay close to your horse's neck, some deer aren't very scared of the horse and don't realize you're there too. Watched twin fox kits play for 45 minutes, and seen some beautiful scenery, flowers and met some nice folks along the way both on horses, walking, biking or out in their yards near a trail.
Just a note to those going off trail (and I do it at times too), be careful and try to "leave no trace". We as trail riders MUST be responsible and take care of the land so that parks, private landowners or whoever "owns" the land doesn't decide we're tearing things up and close the trails. Don't tear up ground and if you carried it in (drinks, food, etc) carry it back out.
If you ride where bikes share trails, try and educate politely even when the rider may be a jerk. Bike riders can generally generate a lot larger number of supporters if it becomes a "horse use vs. bike use" fight . I've found most bike riders to be very respectful of horses and those who don't know best safety policies to be receptive to educating them.