She is cute and your position in the saddle is good. You look like you two will be a really nice pair in time.
I am glad to hear that you are not in any hurry to get where you want to get. So, with that attitude you'll get there sooner than if you try to rush it.
Could be that the horse kind of gets "stuck" when she is asked to move her feet faster . She probably is still working out how to balance herself with a rider on top. We humans don't really give enough "weight" to how much balance matters to horses. As prey animals, they are extremely concerned about keeping their feet and anything that threatens their balance is worrysome. Now, of course you aren't asking the world of her, but maybe she just hasn't realized how to move freely with the weight of a human on top.
If you ask for a trot and she seems "stuck", you might try turning her and walking off, then asking again.
I think I would do a lot more work in the round pen having her change directions, go up and down the gaits (into canter , too).
Will she follow the leadline pressure well? Lots of work getting her soft to the leadrope will soften her attitude also.