When the Gemara answers that there were 3 periods of 2 months each, why doesn't the Gemara answer that there were indeed 3 periods of 3 months, but that the first be'ilah occurred on the first night of the na'arut, the second be'ilah occurred 3 months later and the last be'ilah occurred on the last night of the na'arut?
Technically speaking, you are correct; the Gemara could have answered that there were 3 periods of 3 months, and the Be'ilos occurred in exactly the sequence that you suggest (the first Be'ilah at the beginning of Na'arus, the second three months later, and the third at the end of Na'arus). TOSFOS (DH Ela) in fact asks your question. In order to understand his answer, let us review some points in the Sugya.
Firstly, let's not forget that Rav discussed a case where there were not three, but four Be'ilos altogether. Secondly, let us remember that Rav was discussing a situation where there was exactly a 12 month span between the first and last Be'ilos. The Gemara's answer of 3 2-month periods only accounts for the last three of the 4 Be'ilos, and for 6 out of 12 months. The fourth Be'ilah was 6 months before the beginning of Na'arus, making a total of 12 months between the first and last Be'ilos.
Now for your question: why didn't the Gemara explain that the second Be'ilah was at the beginning of Na'arus, instead of 2 months into Na'arus?
Tosfos suggests a two-part answer. (1) The Gemara prefers to find a case where there are equal time spans between the 4 Be'ilos. (2) Since all Be'ilos of Katnus are only counted as one, the man will not have to abstain from his wife at all when she is a Ketanah. Therefore, it should be assumed that the last Be'ilah of Katnus occurred immediately before Na'arus.
If so, had the second Be'ilah (i.e., the first one of Na'arus) occurred immediately at the onset of Na'arus, it would be occurring just one day before the second Be'ilah, and not at the same interval as the following Be'ilos. The Gemara therefore preferred to give an answer that put all four Be'ilos at precisely equal intervals. It did so by having the first Be'ilah of Na'arus occur after two months of Na'arus. This way, we can still fit in two more Be'ilos during the age of Na'arus (which lasts 6 months), and have each of them separated from the previous one by exactly two months (that is, one at the end of Katnus, the second two months into Na'arus, the third two months later or four months into Na'arus, and the fourth Be'ilah two months after that, at the end of Na'arus.)
This is the answer that Tosfos (DH Ela) gives to your question.
Be well,
-Mordecai Kornfeld