Names are often one of the first labels we get. It often makes us who we are and is deeply embedded in our sense of self. It's the word the world knows you as. Many people change their names to make them feel more like themselves. This speaks volumes to the effect a name has on you.
A name is a word like any other, just used to label someone. Like any other word, saying the word properly is important, especially because it is tied so closely to someone's sense of self.
Butchering someone's name consistently is simply unacceptable. Sure some names may be harder to pronounce and may seem unnatural but not trying is not okay. If you can get names like Stravinsky and Chmerkovskiy, a Nandini, Radhika, or Namrata shouldn't be too hard.
For some reason, it often seems like people have a hard time pronouncing names of people of color, which honestly seems a little odd to me considering many caucasian names are just as unconventional.
Comedian Hasan Minhaj recently appeared on "The Ellen DeGeneres Show" and addressed this issue. He pointed out that no one has an issue with the name Ansel Elgort, but they seem to have an issue with his name. Later, he joked that he goes by the name Timothée Chalamet at Starbucks, which they can handle just fine.
Mistakes are okay. We're all human and you're not expected to get everything correct on the first try. But simply accepting that you can't say it and not making an effort is disrespectful.
You don't need to say the name in the accent of the culture it's from. Say it in your accent but like any other word, keep the same number of syllables and put an emphasis on the correct vowels. Eventually, getting it will expand your horizons just a little bit more. Either way, trying is better than not trying.
That being said, to the people that need to correct others' pronunciation on their name, do it. If you don't tell people how to pronounce it, you can't expect them to get it properly. Many of us introduce ourselves with a name that isn't truly ours; it's a version that accommodates others.
Remember though, there is no shame in your beautiful, unique name so there is no need to anglicize it. Your parents didn't give you that name for it to fit better in other people's mouths.
Our names make up a large part of our basic identity and getting it right is beneficial for everyone.