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Well said, my late dad was always very good with the shovel on the farm.
After being liberal for some time spotted on the kitchen top working surface decided thats it. I glue trapped one in my old place between the dual aspect room central doors running back into the other room. Straight out smashed it against the front garden wall. This was in the evening; nothing left in the morning.
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You don't even need the argument from illnesses. Mice are wild animals and if they get into houses there's a culture clash.
You don't have to 'rebuild' houses to keep the mice out. If people spent a fraction of what they spend on 'pest control' on closing holes effectively (that is, not just superficially) it would be prevention for years. The problem is that it's evidently not as lucrative a business as 'pest control' (i.e., take ineffective and cruel action only for the problem to the guaranteed to return very quickly = kerching). Often it only takes a simple job with a sealant gun with a long nozzle. They will take a while to get back in through that.
I do agree that glue traps are horrible and I think if you do use them you have to be prepared to finish the mouse off quickly as soon as you find it - stomping on it's head hard will do it.
I don't agree that trapping mice makes no sense - it's impossible to close up all the holes in a lot of older house without rebuilding them as @Sharkstar says.
Mice are a health risk, you don't want to be sharing your living space with something that can carry hantavirus, salmonella or lyme disease.