Is your daughter the youngest of the three? I also have a seven year old who has tried various activities with limited success. She is the baby of the family. Part of me is convinced that she has to "fight" for her place in the family. . . and has been dragged to various places to support her siblings for so long that being at home and having her own down time is special. She also seems to crave time alone with either my husband or myself.
She, too, loves to swim, but won't be convinced to consider the swim the team. She has tried baseball, soccer, gymnastics, etc.
We have found that she seems to enjoy activities that keep all of those involved on a similar level with little competition. Even with art, some students do better than others.
My daughter has thoroughly enjoyed getting involved with the Cloverbud 4H program in our community. They meet once each month and do activities ranging from games to field trip to cooking and crafts. It is more social than competitive, and she has a great time. We are also looking into Brownies--as the scouting program follows a similar path. Another option--depending on your family's beliefs-- would be church children's programs. My daughter has also enjoyed such programs as they seem to follow a similar format to the other activities in which she thrives.
I wouldn't push the other activities. Instead, cultivate her interests. Look for programs that make her comfortable. Do an activity together as a family in a low-key way--such as bowling or a nature hike or bicycling. If she enjoys them, it might lead to something she will want to do on her own. Eventually she will find the one that makes her happy. Also, if she has a special friend that is involved in certain activities, that might add to the enjoyment.
There has actually been a lot of research lately that encourages more "play" time for children--stressing that time spent in "organized" activities actually decreases the creative side of their personalities. . . and that too often, more time is spent driving than actually engaged in the activity. Including sports and active pursuits as part of your family life seems to not only improve everyone's health, but also leads to interests that your children can follow thorughout their lives. I think that if you stay active as a family and let your daughter's personality shine, she will lead all of you toward the things that make her happy.