There are 3 major reasons - lack of time, lack of knowledge, and lack of motivation. Let's take a closer look at each of them.
Lack of time. Here I don't mean that a person doesn't have time to practice. No, instead he/she would play so many new pieces that looking deeply at some of them is too much trouble. Of course, some people don't even find enough time to play their pieces, not to mention doing careful analysis. When they do practice, they usually only play the pieces they are working on.
Lack of knowledge. Obviously that's a big issue. In order to examine the chords, harmonies and modulations, you have to have a certain training in music theory. Not necessarily in advanced music theory but some basic skills. I think that a lot of people simply wouldn't know how to do it (curiously, not even every music teacher has adequate training in music theory).
Lack of motivation. If you are to do something as obscure as analyzing chords in your organ piece, you certainly have to have enough motivation to do it. A proper motivation comes from seeing the value of chordal analysis which in person's mind has to be greater than the trouble of this procedure.
We as teachers sometimes are guilty of making the process of analysis look too difficult and too intimidating to the student. Sometimes the assignments are too dry and theoretical with little practical application which may take away all the motivation and curiosity from the student for years to come.
Because very few people choose to go through the trouble of examining each chord, you would be in an advantageous position, if you would do it. Precisely because of that you could become a person whom your peers could look up to. Your organ playing will change, too, because you will start to think like a composer.