Like you, I have a hard time getting something to work properly without a deep knowledge of exactly how it works. My driving teacher spent hours repeatedly explaining the sequence of steps required to drive a manual transmission and I just couldn't get it. Later, when I looked up the internal mechanics of exactly how a manual transmission works, it immediately clicked and I could suddenly drive a manual without issue.
So I too have an insatiable need to understand exactly what (and why) I am playing in order to play it properly. When I decide to learn a new piece, I carefully analyze each chord and passage, trying to internalize the technical structure and decipher the composers original intent. I also take time to gain an understanding of why exactly the composer chose the key that they did; oftentimes this also involves some study of why they didn't chose certain other keys.
This often comes down, largely, to fingering and technique. As you get into more advanced pieces, certain keys just don't make sense and may be nearly impossible to play. At the end of the day, classical composers had a very deep understanding of music and their choice of key was not arbitrary. Figuring out why a piece is in a particular key may well give you more insight and understanding than figuring out how to play it in other keys.
Now, to actually answer your question: why are piano students taught to only play songs in one key?
The main point of piano lessons is to learn proper playing technique, not music theory. Naturally, some theory must be taught, but it is usually secondary and supportive. If additional knowledge of theory is desired, one must seek out other sources, like online tutorials, music theory dictionaries, college courses, or this site (obviously).
(Alternatively, you can do what I did and try to find a teacher that loves theory as much as you do so you can lure them into spending the entire hour session in a deep theory dive! That is, until your parents find out that you have been spending most of your lessons chatting and not actually playing anything, so they get mad at your teacher who explains to them that you are "advancing" into jazz improvisation for which you need a solid foundation of music theory, after which your sessions are split between discussing theory and screwing around on the piano...yup, he was by far my favorite teacher!)
The ability to transpose, either in advance or on the fly, is a practical application of music theory, not a physical skill. A steady progression of the student's skill and ability is the closest thing a piano teacher has to a deliverable "product", and is therefor the easiest way to demonstrate to the student (or their parents) that they are getting their moneys worth.
Lastly, as alluded to before, most pieces, especially more advanced ones, are intentionally written in a key that provides a stable structure for proper technique. For example, pieces that require the thumb to frequently pass under the other fingers are often written in keys with lots of sharps/flats to make this motion easier and more fluid. Practicing it in a different key can make it more awkward to play, and can even result in picking up bad habits, negatively impacting the advancement of technical skill, thereby undermining the very reason for taking lessons in the first place.