Sort of. I’d speculate that those who care about it the most are people who’ve never struggled and failed to pronounce a name in another language. In my limited experience of knowing 3 languages I never had trouble with any pronunciation. So I came to the unfounded conclusion that a. I can pronounce anything and b. everyone should pronounce everything correctly and if they don’t, it’s for lack of effort.
I was wrong. I learned this while learning how to pronounce Chinese names. I thought I was saying them right. For example I’d pronounce “Xi” as “Shi”. But it wasn’t the right Shi, evidently (https://en.m.wikipedia.org/wiki/Lion-Eating_Poet_in_the_Ston...). My friend sat with me patiently as I attempted it over and over. Every time she’d say “no, not Shi, it’s more like Shi”. I couldn’t hear the difference. So even if I put in more time, I wouldn’t succeed.
So now I’m more tolerant of people making mistakes. As long as they make an effort, it’s good enough.
If anyone would like a challenge, try pronouncing the word “Tamil” from the language Tamil. I haven’t heard anyone who wasn’t a native speaker pronounce it correctly.