Ah, yes. That straight man we ordered has arrived right on schedule.
Cogsworth is the Beast's majordomo: the chief servant of the household and the guy in charge of all the other servants. That probably explains why he is such a bossy little control freak most of the time and why the spell turned him into a (very tightly wound) clock instead of something a little more laid-back.
He runs the household well enough even though nobody takes his little temper tantrums seriously, and he seems to end up with pie on his face a lot. He's all stuffy and humorless and just waiting to have the steam taken out of his stride. "Couldn't keep quiet, could we?" he snarks at Lumiere. "Just had to invite him to stay, didn't we? Serve him tea, sit in the master's chair, pet the pooch…" Clearly, the staff has seen these snits before.
And, unfortunately for Cogsworth, they're not exactly paying attention anymore.
The good news is that he still seems to be able to more or less direct the gang. They're gonna give him a hard time, but at the end of the day, they like him. Witness Lumiere, with whom he bickers constantly and who usually ends up getting the last laugh. Some of that comes from the natural nationalistic rivalry of the two characters (Lumiere is French, Cogsworth clearly British), but the rest of it stems from one guy who wants everyone to have a good time, and one guy who worries he didn't do more to prevent it.
Like most straight men, he ends up having a heart of gold. Watch the way he leaps to Lumiere's rescue during the final battle or his instant response when the villagers are marching on the castle. He may be a dolt, but he's a loyal dolt; when push comes to shove, he'll stand with his buddies rather than run and hide.
Cogsworth is a jerk sometimes, but he's our jerk, and in a world with guys like Gaston running around, that makes a big difference.