Avi,
It seems it was a source of veto power for members of Sanhedrin
Choose not mean as in the western Popular Democratic sense of voting choice
1.They were supposed to have all the requisite qualifications
2 If a majority of the people backed Sadducees for example
the leaders eg Shimon ben Shetach were Entitled and obligated still
to throw them out
3.According to the braisa
they were promoted from lower Courts based on pure character ,
meritocratic ,And perhaps dialectic traits