Yes I don't think this thread is the right place to debate whether or not it's ever right to sell a pony. It's a fair and interesting question that I could give my own answer to but if you want to hear or give your opinion on that could I respectfully ask you to start a different thread.
The question OP actually asked was whether this pony can reasonably/fairly do the job of being her DD's all rounder and competition horse. Whether or not the pony is retired to be a field ornament, sold, loaned, kept as a hack, whatever, doesn't really change the answer to that question.
In addition to what I've already said on that, I would add that I think whoever said the head-shaking behaviour could be a response to competition nerves in the rider has a good point. Not to blame the DD or say she isn't a good rider or even is consciously anxious or doing anything different at all - but if the mare goes well and doesn't headshake with all other variables being the same i.e. over the same jumps at the same venue, in the same conditions i.e. without other horses or warming up with others then going away from them into the arena alone (which can be a really big anxiety trigger for lots of horses) then you have to look at the variable that is changing, which is that the rider has the pressure of knowing it's a competition which always will change something, even if totally imperceptible and subconscious. Sadly if it is that, my experience is it can be hard to fix once the pattern sets in, as if the rider is expecting the pony to not go well in the competition that makes them more nervous and tense, which makes the pony more nervous, in turn making the rider even more nervous and so on. And of course even if the rider is 100% totally relaxed and confident there is still the inherant / intangible competition atmosphere, other nervy horses and rider around, bad behaviour in the warm-up etc which it's impossible to totally avoid. Lots and lots of very low key outings can help, sometimes not even getting on or just warming up and not actually going in to the arena, or doing clear round SJ where you just go in, pop some easy fences then pat the pony and go home, as of course does addressing any physical problems. But comes a point where if the pony just is sensitive to the competition atmosphere it may never be something comfortable for them - is it fair then to either pony or rider to keep going and pushing them to compete? I think not personally.
As I say what you then do with the pony is a seperate question and I don't think questioning the OP's morals and accusing her of cruelty for even thinking of selling is helpful...