There is a reason why London has the highest social mobility in the country. My Dh couldnt afford bus fare as a teenager and went to a school where they didnt teach maths or english (ultra orthodox jewish primary school) but he was able to cycle to a jewish secondary (mainstream one which is the top comp in the country). Today he would travel free on the bus.
When he got into a london university, he was able to live at home. Same during his gap year where he earned and saved 10k, he commuted to a college in notting hill where he worked. We married in our early 20s and i moved into his mum's house, we saved 60k in 3 years and bought our 2 bed flat in our 20s, dh has worked for an investment bank since 2017. It was just as well as that he worked for a bank as it was through the bank's private healthcare that we discovered he had functional neurological disorder probably caused from childhood neglect/trauma. It took probably 40k of medical expenses he claimed from the private healthcare system to find that out (and in his 30s), nhs didnt do anything for years. They misdiagnosed it as epilepsy and advised he shouldnt learn to drive.
I understand now why dh was so insistent we stay in zone 3 london with our 400k budget in 2019. I am an immigrant so all i have known is London but for him as someone who had no primary education, health issues he had to battle through, what he needed wasnt a house with garden. What he needed was basic scaffolding- public transport to places that could change his destiny, access to the best healthcare specialists in the country (unfortunately you get that through private healthcare). Due to the breakdown in the uk social contract, the only place that offers this is London because 70% of highly skilled jobs are in London and it is these jobs that offer you healthcare which you need if you grew up poor and suffered abuse, the after effects will be suffered for the rest of our lives.
Now we have a son, i am also grateful that we can send him to London's great primary schools. There is a c of e school that we can get to via bys where 56% of kids perform a higher level in english and maths (compared to 8% in the country)and where just attending church would get us in even as practising progressive Jews. And as a teen he can go to jewish school.
I dont think my husband could do what he did in the north. He probably would have been forced to emigrate or be stuck in mimimum wage jobs all his life as his condition means its not safe for him to drive. While in London he could buy within 3 years of graduation. We probably wouldnt move out of our 2 bed flat and are sticking to one child but healthcare and education and mobility is more important to us than square footage. If you are sick, being able to book a GP appointment and get seen in a hour, getting a specialist referral within a week is lifechanging. So many people are in my DH's position and its probably higher for poor people, they should stay in london if there is a chance they can get these benefits because the nhs is not going to help them. At leaat not on its own, they need nhs and private.