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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.
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Our flat is in the beer cellar of a victorian pub. The mice are getting in from one or more places behind the kitchen units, most likely where the pipes come in and out from the street. I've looked at it and I would have to dismantle at least two or three of the units in order to get to the ingress points and seal them up. It's a big job, and there's always the risk that it unravels somehow halfway through and it takes forever or the moneygun has to be deployed. I've sealed up everywhere else.
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.