While I was reading your post I was thinking, one way is you sit at the piano and you play some of your own note ideas,I have no good idea what could be meant be "understanding a piece".
Grasping the structure, the direction, the technique, the craftmanship, ...? What do you know about the Winterreise or the finale of Mahler #9, if you understood all these technical details? Let me call Mahler as a witness, who said: "“What is best in music is not to be found in the notes.”
Getting to know the piece and trying to relate the parts to the whole? I like this one. A whole and single parts and their relations to each other and to the whole. Maybe. But what kind of relation do you mean? Structural? Emotion-based?
Making out patterns and keeping track of what's going on? But how do you know, what's really going on? On which layer - the audible layer, the meta-layer of semantics, the meta-meta-layer of quotations, ...?
Do you think you understand Beethoven's fifth symphony? Or "Für Elise"?
Do you think Karajan understood Beethoven's fifth? And Harnoncourt? If "yes" for both, why does it sound so different when Karajan conducts it and when Harnoncourt conducts it?
I have no good idea what "understanding a piece" could mean.
..and then you play the inspiring notes that impressed you from a classical piece. You hear and feel the differences in effectiveness. A person curious about the fundamentals of music would want to know what the differences are (facts) and why there are such differences coming out of the physics and interacting with the human brain. We might only think about such a deep subject for a few seconds, but it makes it all very open-ended and never-ending. You can follow this pursuit for many decades.