Whether to kill a mouse?

Modern question of morality considering no-kill and kill -mousetraps. People like Neil deGrasse Tyson have made a claim that a no kill-trap is hypocritical compared to a kill trap.
Neil even said one should instead let the mouse live in their house if they don’t want hurt the mouse. As in the wild the mouse will face a natural death that might not be quick and painless. He went on to then talk about how setting the mouse free instead of killing it to be hypocritical because people cut up trees to make houses.

These points in my eyes reflect an edgy teenager level argument to the subject. As firstly people cut up trees to make houses as people tend to live better lives when they have a house. So people cutting trees comes out of necessity whereas using a kill or a no-kill trap doesn’t.

Letting the mouse live in the house.

The goal is to get rid of the mouse from the house. Also usually there is soon many more of them and they’ll spread to your neighbors’ houses. (mice bite electric wires and in some regions carry diseases, so you don’t want them in your house.) So offering “letting the mouse live in the house” as a more moral solution compared to releasing it in the wild, doesn’t actually solve the problem. So it’s a dumb argument to make and carries 0 real life value.

Is it more harmful to the mouse to be set-free in the wild than to be killed right away?

If you only consider the quickness or painlessness of death, you might opt for instantly killing the rodent. As dying by getting cut up by an owl is more painful. But if you consider living a life to have value, then it’s arguably more harmful to end the life than to give the mouse a chance to live longer in it’s natural habitat.

The no-kill trap objectively extends mouse’s life and even as it leads to most likely a more violent death (kill traps can also fail and cause immense and long pain before the mouse dies), I’d argue most people would also choose for themselves a chance to live longer than to instantly die. And I see very few advocating for nuclear annihilation of human kind as a solution to human suffering, because slowly improving things is “hypocritical” compared to having an utopia.

In conclusion the object of “not hurting the mouse” is achieved by using a no-kill trap. You are not hurting the mouse any more than seeing a completely wild mouse in the field.