I think he's cute!! The pictures aren't the best quality so I'm hesitant to say much, but I love Appendix crosses and he seems to be put together nicely, however out of shape as you have mentioned. Looks to have good, solid bone and I like his legs, but he does seem to be built down-hill. Maybe his neck ties in low? I'm bad at judging that.
As for the videos... I see a horse who needs to be taught to reach under himself a little more (I feel like he strides a little short in the back, but that could be due to the way he is being ridden), and also a horse who needs to learn to accept light contact. I don't love the way your trainer is riding him (but who am I to judge!).
What kind of bit is he in? If I were in your shoes and he was already in a soft bit (a plain or double-joint snaffle), I would start asking him to move out with big strides (use your legs to really get his butt moving!) and offering little-to-no contact the reins--instead asking him to relax his neck and move forward. Over time, you can offer a light contact and push him into that contact with your legs, but for now I feel like he is so focusing on resisting bit contact that he doesn't move out in an effective way. You need to gain his trust in your hands before you can offer that consistent contact.
That's my two cents! I'm no professional, so take it with a grain of salt.
ETA... (and not related to confo!) in the video, it seems his saddle is a hint far forward which might have to do with his resistance. Make sure that you take care to feel where the shoulder is, and not to place the saddle on top of it! (It should be just behind it so that you aren't hindering his movement.)