[Avodah] Machlokes in Mishnayos, why?

[Marty Bluke via Avodah]

The Rambam in Hilchos Mamrim (1:4) writes:
"When the Beis Din Hagadol was in existence there was no machlokes in
Israel
...
They would come to the Beis Din at the entrance to the Azara. If they knew
[the din] they would tell them, if not everyone went into the Beis Din and
asked.
...
If the matter was not clear to the Beis Din they would discuss the issue
until all agreed or they they would take a vote and follow the majority and
tell the petitioners this is the halacha"

The Rambam is based on the Gemara in Sanhedrin 88 which states the
following:
"At first there were no arguments. Any question would be brought to the
local Sanhedrin. If they couldn't answer, it would be brought to the
Sanhedriyos outside the Mikdash, and if needed, to the Beis Din Hagadol.
The Beis Din Hagadol was in Lishkas ha'Gazis from (the time of) the morning
Tamid until the afternoon Tamid. On Shabbos and Yom Tov, they would sit in
the Cheil (just outside the Azarah). If they had no tradition about the
matter, they would vote to decide. After there were many Talmidim of Hillel
and Shamai that did not learn enough from their Rebbeyim, many arguments
arose, as if there were two different Toros in Yisrael (Beis Hillel and
Beis Shamai)."

There are 2 obvious questions on the Rambam (and really the Gemara):
1. The Beis Din Hagadol was in existence throughout the period of the
Tannaim, the Gemara in fact related that they left their place in the
Lishkas ha'Gazis 40 years before the destruction of the Beis Hamikdash. If
so how can the Rambam write When the Beis Din Hagadol was in existence
there was no machlokes in Israel when Hillel and Shammai and their students
had disputes that were unresolved when there clearly was a Beis Din Hagadol
in existence?
2. Why didn't the Beis Din Hagadol resolve the disputes between Hillel and
Shammai and their students and in fact all of the disputes in the Mishna?
Why did they let Machlokes fester? Their clearly was a Beis Din Hagadol for
most if not all of the period of the Tannaim.
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