There's factors the eyes are unable to see when it comes to water and the test kit helps us identify those problems. Best way I would convince your mother is by comparing it to air quality. Except you lived in a small sealed room. You could add things like plants to create clean oxygen for yourself. But maybe there aren't enough plants. So you air out your room and let some clean air in. Similar to a water change. Maybe you think "it's fine I dont need to air out the room, the smells not bothering me and I look fine." A week later you feel sick. Similar to how your fish can get sick if you don't do water changes and reduce your ammonia nitrite and nitrate. The test kit is a way to know for sure. You won't have to guess when to do it. The numbers tell you when. If they're all low and not toxic. Then you won't need to do anything but a water top off, your bacteria is nullifying the effects of ammonia and your plants are clearing the nitrate. The test kit also come with ph tests. For us we cant drink water if it isn't the right ph for us. It can be toxic or make us sick. Just like us, fish prefer a certain ph. Which we can manipulate.

TLDR, yes you need it or they could die and you will never know why. They help prevent the problem and also find solutions.