A local friend of mine used to say he wished it would get colder here to kill off the mosquitoes. I had to point out to him that some of the worst areas for mosquito infestations are within the arctic circle. Many reviews I have read of people going on trips to Alaska commented on the horrendous swarms of mosquitoes up there. Yeah, you wouldn't have them if it stayed below zero all year long, but that would not be a suitable trade off for me.
Maybe the cold will kill off some insects, but it isn't a complete solution to that problem. Eggs will be laid during the active season, and those eggs will hatch when it warms up again. If cold snaps were a solution to the insect problem, then northern states would be completely and permanently devoid of insects, which certainly is not the case.
As for weeds, heck, they would be back in the Spring, just like the "good" plants that would get kicked back. Nature is just like that.
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