I don't think the bullpen is the problem though. And it wasn't last year either. Last year it was the rotation (on a team that didn't have a good offense). This year it's mostly the offense (but it's been a bad week for the rotation too).
And I don't see how the manager is supposed to use the bullpen differently when half the pen is on the DL and the starters are only going four or five innings. There are certainly times Gabe overmanages the pen in close games but today wasn't one of those. The only solution to overuse is to use more guys you don't actually want to see in higher-leverage situations. Or overuse the few guys you do.
The pen will work itself out or be addressed in July, but when your pre-season 1-3 starting pitchers are all faltering and Bryce Harper, Maikel Franco and JT Realmuto are getting out-slugged by Cesar Hernandez that's not a good formula for success.
Another part of the story (besides the injuries) is that without the huge top-of-the-lineup production Franco is no longer a threat (and is in fact a negative-WAR player) while Odubel really hasn't been all year (him too).
Also, defense. "Better than last year" is not actually "good."