I don’t want to sound native! Why does having an accent matter?

Unless you are a secret agent, having a slight accent is not an issue, as long as you can be well understood. Even having a stronger accent won’t impede communication if the people you talk to are used to listening to people with that specific accent. That is the case with Indian people in the UK, or Mexican people in certain areas of the US. But the stronger your accent is, the more that native speakers will struggle to understand you. You’ll have to repeat or rephrase more things, they’ll have to use more mental effort to try to understand you, and your interactions will tend to be shorter and more tiring. While you don’t need to get people to believe that you are a native speaker of the language, having a clear pronunciation is always a positive thing, especially when there’s no extra effort required to get it.