Good question. The first verse was dropped, because it was the only verse that was sung in the Third Reich and was so strongly associated with Nazi ideology (also in part due to the way you can interpret it) that it was too uncomfortable as official anthem.
Why we have only the third verse left, has a simple reason: Adenauer said so.
Daher die erneute Bitte der Bundesregierung, das Hoffmann-Haydn'sche Lied als Nationalhymne anzuerkennen. Bei staatlichen Veranstaltungen soll die dritte Strophe gesungen werden.
-- Adenauer, 1954
(emphasis by me)
It is argumentative why he chose the third verse exclusively because according to the sources I found, he didn't care to explain his motivation at any moment.
A cynical person would say, that you can be lucky if the average German manages to learn one verse. But seeing that the whole anthem was going to be replaced, I think it is more reasonable to assume that Adenauer thought he would have better chances asking for one verse to be used than two. Maybe they planned to eventually add new verses?
You might still ask why the third, and not the second. I see two possible explanations;
- The third is further away from the evil first verse.
- He simply liked the third better than the second (not necessarily a question of taste but of politics).
From today's politics, I would say that the second verse would be considered offensive to some people due to the whole gender discussion ("Deutsche Frauen, ..." might be considered objectifying in that context - I don't know, just guessing).
Finally, one could argue that "Einigkeit und Recht und Freiheit" are simply some of the most incorruptible virtues you can get. A good way to get rid of the fascist NS image.