The red boxes are more orientated around ILS approaches (they won’t match the ILS exactly, but will be good enough to get you in at a suitable rate). Some airports, such as this one, will utilise a visual approach as terrain may not permit it to use an ILS approach. This specific approach into Kai Tak is known as the Checkerboard Mountain Approach, which passes over central Hong Kong and turns sharply onto short final to land.
Here’s a video if you’ve never seen it before:
It will take a few tries to get the hang of it, but overall it is a very exciting approach and I would definitely recommend it!
As for seeing airports through mountains, this issue has been around for a while. Hopefully it’ll be fixed in the future.