Appreciation for Shostakovich Piano Concerto No. 2 - second movement - Andante. What makes it sound so particular?

Excuse my rambling. This is more an appreciation discussion, and a question for similar pieces. The chords, their progression, the accompaniment, everything together makes me cry almost every time I hear it. The best recording of it even features his own son and grandson performing, which makes it even more special and emotional, considering it was written for his son. https://youtu.be/z3wmDrGUFeM?si=nULrKrNNmrr55ob7 (I’m on mobile without app so cannot post an actual link unfortunately).

I really love this piece, and I can’t really pinpoint as to why, in the harmonic sense. To clarify, I studied at the conservatory some ten years ago (wind) but my harmonic and technical insights have since declined. I’m not performing anymore. Can anyone help me analyse or explain the piece, why the chords are simple yet mesmerising? I suspect it’s due to sudden key changes. That’s it?

And this piece is not too difficult for a low intermediate or above beginner player. Not to perform perfectly, but skill wise. Are there any other similar concerto’s that capture this same feeling but are also playable for the non so skilled?