There’s a balance to be struck!
Try and meet her half way op.
Ironically one of the ways to deal with this is to make her feel as though she has more control which is very difficult when you are trying to get her to do things she doesn’t particularly want to do.
It works better if the activity is her idea, so you have to try and subtly facilitate that. Sometimes doing activities with friends takes the pressure off a bit, so the friendship becomes the point, not the sport.
Compromise on a physical activity she loves: dancing, bicycling, horse riding, climbing but completely back off in terms of pressure.
Maybe go bicycling with her regularly and take her for a treat afterwards or take a picnic? Show her that sport can be fun. School PE can be so off-putting!
And if she enjoys being at home, then 10 is the perfect age to introduce creative activities like sewing, felting, knitting, baking, gardening drawing etc as a substitute for some of the tv she is watching. Or get a dog she can walk? And obviously set time limits if she has a phone. Apps like Instagram are made to be addictive.
I agree with pp that tv only becomes an issue if she is using it to avoid things due to anxiety or depression. And yes, dc can get depressed.
Maybe look in to getting her some help with her anxiety now so it doesn’t develop further and in order to develop resilience all you have to do really is praise effort not results and completely back off and model it!
I got travel sick as a child so I really sympathise with that but again you need to check she is not using that as an excuse to avoid doing things as that is a bad pattern to get in to.
So allow her more control and to be able to speak freely and honestly and say “I don’t want to go to x as I would prefer to be at home” and then you can have a longer non -stressy conversation about whether she feels ok in herself, whether she is tired or anxious, what really lies behind her refusal, and you can say “ok that’s fine but in return I would like it if you would come to with me next week’ and make sure she sticks to it. If she protests again, calmly remind her you had a deal.
One of the most useful skills in life is learning how to make yourself do stuff you don’t want to do. So try and be more of a facilitator than an enforcer.
Some children just feel a lot better if everything that is going to happen is explained in detail to them in advance. You can sometimes sit in on a sports lesson the term before you start to do it, and then you can help your child to familiarise themselves with the journey there and back, where the loos are, how to tie those particular shoe laces etc. Sometimes it’s as simple as that.
If you are resilient op then it’s highly likely that your dc will grow up to be resilient too. But everyone has different genetics, socialisation, capabilities. She might just need a bit more time then you did so try not to be frustrated and give her more responsibility and control over the choices she makes.
I totally understand where you are coming from though op as my dc have had loads more opportunities than I did as a child and they rejected half of them during their late teens. They did carry on with a couple each though and that’s good enough!