I think the bullpen problem has been more with the manager than the bullpen itself. The Cubs are 17th in the majors in bullpen ERA--solidly middle of the pack. They've given up only 3 unearned runs, so it's not like they've hidden their poor performance by giving up a lot of unearned runs (19 teams have given up more unearned runs, just for context). On the whole, they really haven't been that bad. Several teams that are generally considered to be good (Orioles, Nationals, Red Sox, Indians, Cardinals, Rays) have higher bullpen ERAs.
And even if you don't think they're a middle of the pack bullpen (as the stats say), their impact is limited. Cubs relievers have only pitched 134 innings this year...only three bullpens have been used less often. The Cardinals (133 1/3) and the Phillies (133) are within an inning of the Cubs, so there isn't really a difference there. I think it's fair to say the overall bullpen performance isn't helping the Cubs...but I don't think it's fair to say that they have performed poorly enough that they are sabotaging the team.
It's really Sveum's bullpen usage that is the problem. His depth chart seems to have Russell listed about 4th or 5th (even though he has the fifth highest WAR among major league relievers so far this year), and seems to have Marmol and Dolis as potential setup guys. In all the time the Cubs were struggling for a closer before Gregg showed up, the only guy who didn't get a chance to close out a game was Russell. Camp was pitching in (and blowing) high leverage situations for a month with an injury, and Sveum never even seemed to consider demoting him. He looks at Villanueva as a one inning guy...he was even quoted as saying that Villanueva was too good to be "just" a long reliever (as if pitching 2-3 innings in a close game isn't more valuable than pitching 1 inning). Failing to piggyback him with Garza in that game in Pittsburgh was awful.
It's not the bullpen that is the problem...it's the guy in the dugout.