Personally I wouldn’t and didn’t.
At work I encounter hundreds of people on phone/email and never meet them. The names I remember most easily are the familiar ones.
If someone says “speak to x or y” I’ll usually scribble down the easiest one to hear/remember/spell first and look them up (even if I have to try a few known variations of spelling when I look them up) rather than interrupting the flow of conversation to ask the speaker to repeat the other name, and then to separate out which bit of what they said is the first name, and which bit is the surname, and then ask them to spell it, which they might need to look it up themselves to do.
Maybe not everyone has this but I find it hard to even ‘hear‘ a name sometimes, I remember having to ask someone’s name 3 times and then give up as although I could hear her perfectly and I wanted to be able to repeat the name back to them, I just couldn’t decode the sounds in my mind (I’ve tried to remember that person’s name just now for example, it was a Scottish name, but I can’t remember it even I knew her for 4 years).
Also thinking about it, even for my kids at nursery, when I try to talk to them about their friends I will remember his friend ‘Thomas’ and say ‘did you play with Thomas today?’ but there’s one little boy he is very friendly with and his name just constantly escapes me even though I try really hard to imprint it in my mind every time I hear it (the carers aren’t supposed to mention other kids names, so I just hear it in passing from time to time). My child can’t pronounce his name properly either and calls him something vague which I can’t copy, so we couldn’t really talk about him properly over lockdown which was frustrating. Same with the carers’ names actually. If I’m not in a position to write the name down straight away after introduction, I find it hard to recall later if it’s a name I’ve not heard before, especially with the kids jumping over me when I’m trying to imprint it.
I think all these little things add up as a disadvantage for people with tricky names.
Maybe kids growing up in London will be different as their ears will grow up familiar with loads of different types of names.