No, these quests do not end. These are simply due to the Radiant A.I., and never ending quests. Also, as the Harbinger, you're not the leader, it would be better described as an adviser to the council of Companions.

What happens at the end of the Companions questline?

At the end of the Companions questline, the last fragments are collected, and Wuuthrad is reforged to become whole. Wuuthrad can be kept after the final quest.

How many quests do I have to do for the Companions?

The primary Companions quest line consists of six quests; multiple radiant quests must also be completed over the course of the quest line. Three achievements (50 points; 2 Bronze and 1 Silver) are unlocked by the Companions quests.