what is the main element of the 团体赛 ? mainly is more for competition than 联谊(=联络友谊) ?

Personally,

两台制 is more for 联谊, examples are Albert Spquare 团体邀请赛 and Pasir Ris East C.C.团体邀请赛


六台制 and 四台制 are more for competition, examples are 文礼全国团体邀请赛 and 狮城杯.