The abundance of one species does not a healthy ecosystem make. I have a friend whose family owns a 1700 acre island off the coast of New England. It used to support an enormous white tail deer population -- and not coincidentally it had a plague of ticks, because everything in nature is food for something else. You would not have wanted to visit there back in the 1970s because the tick problem was insane. Everyone in his family has had Lyme disease, which also feasted on the swollen deer population.
Then in the 1980s the Western Coyote made it to New England, and a pack swam out to the island. In a single season they took down most of the deer herd, and now the island is a pleasant and sanitary place to live. And this is not some kind of odd aberration; this is how ecology works. If you disturb an ecosystem (say by killing off all the native timber wolves), weed species take over and they end up riddled with disease.
Weed species the ones who by sheer luck can live in conjunction with or off of large human populations. In a healthy ecosystem they may be cute, but an ecosystem dominated by weed animals can be nightmarish. I know lots of natural science geeks, and for the most part animals don't scare them. I once went for a walk with a girl who picked up a rotting coyote head and put it in her jacket pocket. She was TA'ing an anatomy course and wanted to show it to her students. But even she wouldn't go near a racoon, because unchecked by predation suburban raccoons are chock full of leptospirosis, salmonella and roundworm -- not to mention rabies. Those diseases can and do cripple, even kill people.
A world dominated by weed species would be quite horrible to live in.