From the Azkaban chapters:

From what Amelia heard, Dumbledore had gotten smarter toward the end of the war, mostly due to Mad-Eye's nonstop nagging; but had relapsed into his foolish mercies the instant Voldemort's body was found.

Dumbledore's lesson from his room isn't that you needed to shut up and multiply, it's that war is so terrible that you must be willing to sacrifice anything so prevent it from occurring again. He prioritized people's lives to stop a war, but he's not willing to sacrifice anyone except to prevent more violence. Dumbledore never wanted to sacrifice his sacred values for the greater good, he was forced to by the war. From "Taboo Tradeoffs":

He had to choose between losing his war and his brother. Albus Dumbledore knows, he learned in the *worst possible way, that there are limits to the value of one life; and it almost broke his sanity to admit it. *

In "Pretending to Be Wise", Dumbledore says that the reason he doesn't subscribe to purely utilitarian ethics is because he doesn't trust himself:

"Grindelwald was my dark mirror, the man I could so easily have been, had I given in to the temptation to believe that I >was a good person, and therefore always in the right. For the greater good, that was his slogan; and he truly believed >it himself, even as he tore at all Europe like a wounded animal."

So he sticks to his virtue ethics, unless he is forced to, since he doesn't trust his morality enough to do non-virtuous things in service of it, lest he become another Grindelwald. It is only when forced to that he abandons his principles, and only to prevent further violence. Choosing to sacrifice someone is against his nature, his room might remind him of the costs of that course of action, but it doesn't change who he is, only make him regret his failure in the War.

Add to that the fact that Filch is someone Dumbledore feels much sympathy toward, and the fact that he wasn't facing Lucius or anyone on the other side, him taking Filch's side is understandable, if not expected.