The sefer Halichos Olam 4:16 by the 15th century Rishon Rav Yeshua ben Yosef HaLevi, coming from the Ramban, says that there are two types of gezeiros shavos. One is that Chazal accepted that two parshiyos are to be learned one from the other, but they weren't sure which words connected the two. One example is the parsha of ones (a rapist) and mefateh (a seducer). The two share similar laws and are learned one from the other. Nevertheless, the gemarra has a dispute about which words connect the two.
Another type of gezeira shava is when Chazal had accepted that a certain word is to be used for a gezeira shava, but they don't know where to apply it. I believe your gemarra is an example. The gemarra is suggesting that the gezeira shava is this particular word (אות, דורות, etc.). However, Chazal didn't know where to apply it. The gemarra is suggesting to apply it in contexts 1 & 2 (like you say), but that's not definitive. If we are to employ such a gezeira shava, perhaps it should be 1 & 3.
In the case of that gemarra for example, it's suggesting to learn Bris Milah overrides Shabbos because of a gezeira shava of אות אות. However, if you were to learn that gezeira shava teaches you that something overrides Shabbos, the gemarra is saying that perhaps the gezeira shava is to apply between Shabbos and Tefillin, which also has such a word. Since we know (from elsewhere) that Tefillin doesn't override Shabbos, the gemarra rejects using that gezeira shava.
I'm not sure if the gemarra is saying to apply it in this context instead, or additionally. Either way, the Halichos Olam and Ramban are limiting how much Chazal really accepted from tradition. The rest was up to them to figure out, where the gemarra you're quoting shows the back and forth.
Why the gemarra doesn't always ask this, I'm not sure. Perhaps finding a case where this happens would make it clear why it didn't.