There are days when I find myself at risk of getting into a battle with my DD over something and I have learned to see that as a warning sign.
My advice is to let it go.
First, let me say that I would be thrilled if my kids stuck to anything for as long as she has done Scouts and Aikido. She has given them both a good go, she has earned the right to quit if she wants to, IMO.
Life is too short to pour your energy into someone else's goals for you. I'm sure she will find something else to get into, maybe not now, maybe right now in her life she needs to turn her energies inwards and grow that way for a bit. Trust her. Trust that you have done right by her and that she will have a fulfilling life even if it doesn't follow your plans for her.
Second, you risk damaging your relationship with her by trying to get her to follow your own plans and wishes for her rather than her own (even if your plans are good ones). Not only do I think this is not healthy for her to follow someone else's idea of what she should be doing, but more importantly it is not worth damaging your relationship with her over these things. They are not life-threatening and will not ruin her future goals and accomplishments.
I can see that being able to defend herself physically is important for you. I am saying this gently: Would you be willing to consider that your own experiences with date-rape may be biasing your judgement? Honestly, I don't think equipping women with martial arts skills is the only solution, and maybe this is more about allowing you to worry less than keeping her safe (and what mama doesn't want to worry less?). She already has more skills and experience in that regard than most women.
Controlling kids is a slippery slope and for me, personally, I hold my relationship with DD over everything. Right now she wants to quit riding and we just started a new term and we almost got into a fight about it when I told her she needed to stick it out to the end of term. I did not let it get to a fight, I let the subject go for now and we will revisit it this week when we are less emotional and together we will come up with some sort of compromise. I will tell her why it's important to me that she stick it through and she will tell me why she wants to quit and we will trust each other to take the other's feelings and wishes into consideration. Eventually we will come up with a solution we can both live with. Not only do I preserve our relationship, but I teach her valuable skills about finding solutions to problems that don't involve wielding power (which is what I am doing when I insist she "has" to do something).
A final word about your "cons". I've already addressed the physical safety stuff. You mentioned college applications. I don't see that as a good reason to damage your relationship with her, either. If she really wants to go to college she will either get in despite quitting or she will find out that she needs to do more extracurricular stuff and build up her application. She may have to wait a year or two to start college - not the end of the world! and she will have learned a valuable lesson that no amount of forcing her to do stuff would ever accomplish. This age is the time to make such mistakes: you have the time to learn from them and try again.
...with all the caveats about this just being my opinion and worth what you paid for it, etc. :)
Homeschooling, Homesteading Mama to DD ('02) and DS ('04)