Open Eyes, I can understand your concern. I really tried to get rid of my depression through other ways before resorting to anti-depressants. (And later I was diagnosed with bipolar disorder.)
Please remember that the list of side effects is a list of things that can happen. Not everybody has these side effects or maybe even any of them. So, we don't know if she will experience problems riding horses until she sees what a particular drug does.
I always encourage people to see a therapist when they are depressed. If the depression is caused by environmental factors and not a brain-chemical problem, then it might be resolved without drugs. However, when someone is getting more and more depressed, even with therapy, exercising, etc., then I think drugs need to be added. (And they do tend to take awhile, such as two weeks, to really kick in!)
However, if your daughter is officially an adult, then her decision will be honored by her health professionals. I know that's frustrating. I find it that way, too.
I have no doubt that such drugs affect brain functioning. That's how they help. If your daughter goes on such a drug, then let's hope she will get better and can wean off of it at some point.