Why is it that, out of the 16 days of non-High Holiday biblical festivals (7 days of Pesach + 1 day of Shavuos + 7 days of Sukkos + 1 day of Shemini Atzeres), only the single day of Shavuos is without neighbors? All the other days are in adjacent calendrical groups. Why is that?
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There are two main Regalim - Pesach and Sukkot. Each one has another one-day mini yom-tov without special mitzvos afterwards. They are each called an "Atzeret" since they have no special mitzvot and are a culmination of the previous holiday. 7 weeks after the beginning of Pesach is Shavuot/Atzeret, and the day after Sukkot is Shemini Atzeret.
The connection between Shemini Atzeres and Sukkos is clear, since there isn't even a separation of days between them. The connection between Pesach and Shavuout can clearly be seen in the the Mitzva of Sefira that connects them and in their connected harvests and korbanot. Also, they left Egypt to receive the Torah on Sinai, which happened on Shavuot.
There are two parallel sets of holidays, so one can then ask why is there a seven-week break between Pesach and its Atzeret, but not between Sukkot and its Atzeret? The Midrash (Yalkut Pinchas #782) asks this question and explains that while the weather is great during shavuot, it would be hard to travel back to Jerusalem during the rainy season, so its Atzeret is right afterwards. (Translation from Sefer Ha-Aggadah):
There are some other comparisons that can be made between the Holidays' sets of korbanot and their harvest seasons that may also add to this explanation.
In addition, while shavuot was when the Jews received the Torah, Sukkot is connected to the celebration after receiving forgiveness and the second set of Luchot on Yom Kippur. So it wouldn't make sense to push Atzeret further away into the year. Perhaps the sefirah-parallel period can instead be seen in the Aseret Yemei haTeshuva (and perhaps Elul) which came before.