The world is globalized, it's not expected as a norm to be able to pronounce every name properly.
My name isn't in "English" and I grew up in the USA. The people whom act like you when I introduce myself and talk are the most annoying. I don't have any onus on everyone pronouncing my name properly and if they say it with English phonetics then that's acceptable because they're trying to adapt their language to my name.
That is much more polite then having this internal monologue and judging/having bias to everyone else that they're "inconsiderate."
Just ask how to pronounce if you can't, read the room and then apologize if it needs to be said.
Loosen up.
There is a very good chance that your correction is coming off as inconsiderate since it is interrupting the flow of conversation and perhaps even the other person's train of thought. That interruption is also going to affect how well they remember the correction. If they were unprepared for an off-topic interjection, it is probable that they didn't hear what was being said or it did not register in their memory.
So yes, getting people's names correct is important. On the other hand, letting people get your name is important (rather than forcing the issue).
To be honest, this seems as one of the worst ways to handle such situations. Feeling distraught over an imprecise pronunciation feels like something to get rid of rather than accepting as a norm.
And I am saying this as someone with a slight speech impediment using one of the more consonant cluster-rich languages on the continent. I would have to be delusional to expect someone from an English-speaking country to pronounce names such as Grzegorz Brzęczyszczykiewicz on first or even second try.
Also, how do you decide which names are common? Not trying to be rude, but this can easily depend on from where someone is coming from and their age/generation. Names popular in the generation of my grandparents sometimes are completely absent from my own and rarely written down.
This all also comes with the prerequisite that a person in question still uses the original pronunciation and not one adapted to the dominant language in the region.
If you want an analogue, think of Muhammad Ali. Arabic name, non-Arabic pronunciation.
That said, I certainly wouldn’t get upset at the effort. I’d assume you mean well, so no foul. It’s just not that important.
It takes me about 5 minutes to Google names I'm unfamiliar with before I start a new class to make sure I get it right. I don't know and I should not be expected to know Chinese, but I think I should be expected to be polite enough to try to get it right before I take a stab and butcher someone's name.
I don't think it is an enormous burden to put on myself, and I don't think I'm performatively self-flagellating by doing it. I think it's just me trying to be polite. I think everyone should try to be polite in as many contexts as possible.
On another context of names:
My own names are Hebrew and Germanic respectively, two syllables each. My given name is common in North America. Most people pronounce it wrong anyway. I would greatly prefer if they didn't. It's a little disrespectful. I politely correct people and then find most of the same people make the same mistake again and again. I find it disrespectful, the same way I find it disrespectful if someone spells my name wrong even though my email address is first@last. It's not because they're frothing at the mouth racists, it's because they're minorly inconsiderate.
My wife has a common name with two English spellings (let's say similar to Meghan/Megan). People frequently misspell it. She's not bothered by people guessing wrong. She is bothered when she corrects them and they still get it wrong. When I met her, it's not like I cosmically knew which of the two spellings was correct. I simply learned which of the two spellings was correct. It was a basic act of politeness and care. When people don't do it, it comes off like they don't care.