Why is Purim celebrated in Adar Sheni?

The Mishnah (Megillah 1:4) states that Megillat Esther is to be read in the second Adar, but the Talmud (Megillah 6b) records an argument concerning this question. R. Elazar says to read the Megillah in the first month of Adar, while R. Shimon ben Gamaliel says to read it in the second month of Adar. The Talmud explains that R. Elazar says to read it in the first month since we do not pass over the opportunity to fulfill commandments, while R. Shimon ben Gamaliel says to read it in the second month to connect the two redemptions, the redemption of Purim with the redemption of Pesach. Of these two arguments, R. Elazar's reason appears to be stronger since we try not to pass over the opportunity to fulfill a commandment. (For some exceptions to this rule, see Mishnah Megillah 1:3 and Talmud Megillah 5a.) Why do we not follow R. Elazar's ruling and read the Megillah and celebrate Purim in the first Adar?

I believe the answer is due to the reading of the four parashiyot: Shekalim, zachor, parah and chodesh, see Mishnah Megillah 3:4 and Talmud, Megillah 29a. The Talmud (Megillah 6b) notes that R. Shimon ben Gamaliel also asserts that the four parashiyot must be read in the second Adar. Adin Steinsaltz (footnotes on his commentary on Talmud Megillah 6b) quotes from the Ritva that it is obvious that the four parashiyot have to be read by the second Adar since three of the four (except zachor) relate to Pesach.

Maybe, the "obviousness" that the four parashiyot are to be read in the second Adar, is the real reason why we read Megillat Esther and celebrate Purim in the second month of Adar. The reading of zachor is the second of the four parashiyot, and it is intrinsically connected with Purim. For example, the Mishnah (Megillah 3:4) notes that zachor cannot be read on Shabbat if the Shabbat is more than one week from Purim. However, zachor is also part of the four parashiyot. 

Accordingly, due to the principle of transitivity, once three of the parashiyot are connected to Pesach and are read in the second month of Adar, then the reading of zachor also must be read in the second Adar. Furthermore, once zachor is read in the second Adar, then as Purim is connected with the reading of zachor, it is also celebrated in the second Adar. Thus, Purim is connected to Pesach through the four parashiyot. This transitivity is strong enough to overcome the argument that we do not pass over the opportunity to fulfill a commandment, and hence the ruling is like R. Shimon ben Gamaliel who maintained that both the four parashiyot and Megillat Esther are to be read in the second month of Adar.  This connection of the four parshiyot to both Purim and Pesach is then the basis for R. Shimon ben Gamaliel’s rationale that the goal is to connect the two redemptions togethers, which is through the four parshiyot.