The White Sox's bullpen was a sliver better than average last year, but the bullpen construction was actually more important than that. The one thing that kept the White Sox from being a mid-70s win team is that they were tied with Cleveland for 4th best record in MLB in 1-run games.
To say that another way, when the White Sox were able to give a tight game to the key 2 or 3 guys in their bullpen, they were commonly able to hold that lead or win that game, at a rate comparable to the best bullpens in baseball. However, the bullpen had very poor depth, so if the middle relievers came into a game, more often than you would ever want - they crumbled.
They do have too much money allocated to the bullpen, but they also have very poor bullpen depth. Trading away Hendriks and moving guys like Bummer and Lambert into key roles in tighter games has a very good chance of making this team suddenly beatable in 1-run games, and thus it has a very good chance of making this team worse.
Take a look at the Phillies and Astros and ask yourself if you have confidence in Bummer, Lambert, Graveman, and Lopez as key contributors in a pennant race or a playoff series? Maybe Lopez is fine, you ready to go to Bummer and Lambert in a key game? We did exactly that in the "must-win" series against Cleveland.