In the second clip your AP goes off and you stop follwing the route, when you turn it back on the aircraft returns to course. In the third clip AP stays on but you havnbot moved the throttles back to CLB as is being suggested in you display. I don;t understand why that would have issues with your flightplan though.
I've not used he default A320 since the free FBW version came out as it allows you to do so much more and is much more reliable.
I would stongly suggest getting this
https://flybywiresim.com/ it's free and the support on disccord is brilliant for new flyers as there are lots of helpful tutorials.