You said stamppotandgravy was making it more complicated than necessary. Not at all.
Imagine a wealthy French woman, living in central London in a big house, paid for by her husband's company, she speaks only French. She speaks to shop assistants and waiters in very broken English. Can't hold a conversation in English. She doesn't work. She has some British cleaners, but they speak GCSE French, so understand her instructions. She has wide and exclusively French, Belgian and Quebecois network of friends plus a few British friends who speak fluent French because they studied there. Her children go to French school in London. And then imagine being told, in French, that she was integrated in British society.
Imagine then a German paediatric nurse. Moves to the London with a young baby and her British partner. He's a British doctor who was working in Germany for a couple of years and speaks fluent German. At home they speak German because her English isn't great. Over the years it gets better, but still full of mistakes. She tries to learn more but she's dyslexic and it's doubly hard. She ends up working in a day care because she can't transfer her nursing qualifications to work in the U.K. purely due to her academic English not being up to scratch. Conversationally it's ok though. Her child goes to English school. He's bilingual. Partner leaves her and takes no responsibility of child, but won't agree to her moving back to Germany. Now she's living on a tiny income, struggling to make ends meet. Her whole life is in English. She's worn down from being a single mother in a different language and having people point out - daily - that she's not British because she's got a heavy accent.
Neither of these is me.
Both of these are real people I know with identifying details replaced with similar/equivalent ones.
Their experience, both as white women living in London, is not remotely similar.
Finally, while you may think that an accent is a sign of something good, you have to remember you're talking about English. We're very used to hearing our language spoken in many different accents and even some mistakes. That's not the case with every language. And when it's not the case, your accent becomes a point of attention, as do even small errors, usually to indicate that you're in some way a bit stupid. It's not with everybody, but you only need a few every day for it to grate. My daycare worker friend had summer camp kids tell her that to her face, with colleagues standing by. It was acceptable because they agreed she clearly wasn't intelligent if she made a single grammatical error!!
Living abroad can be great, in any country. What makes it really hard is lack of money, because that limits your ability to socialise. There can be irritations everywhere, but the isolation faced by someone living abroad whose finances are limited is greater than someone with the same financial limitations in their own country. To need time, money and energy to invest in building a network. Generally money can buy time (don't need to work a job you hate, just to cover bills) and provide a cushion that prevents energy depletion.