I don't think that there is any additional information in the quest journal about Miscellaneous quests (that is, specifically the quests that get filled under the "Miscellaneous" category in the journal). The one line in the journal is all you get.
(Which, yes, is a surprising choice on Bethesda's part. Most RPGs will give you a paragraph or two about any quest in the journal, but Skyrim seemingly doesn't for Miscellaneous quests.)
However, if you've forgotten the context of the quest (which, as you point out, is very easy to do), you can always turn on quest tracking for that individual quest. Make sure that quest tracking is turned on for the Miscellaneous category as a whole (in the left column), and then turn on quest tracking for the individual quest in the right-hand list. (And maybe turn off quest tracking for other quests.)
Also, don't forget that you can select a Miscellaneous quest and get a map location for that quest.
These will let you at least find the person, place, or thing for that quest, even if you can't recall why or what you need to do with them. In almost all cases, what you need to do becomes clear when you get to the person, place, or thing.
My bad. Apparently, I only talked to Aela and Vilkas, and they never had the option to start a new quest. I went back and talked to Farkas and he gave me a new quest.
Best Answer
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.