Unfortunately, many careers are still very much London-based. Once you've established yourself in a profession / industry, it can be hard to move away because it's often easier to change employer and progress your career in London. This is especially important in an age where it's less common to stay in an organisation for many years. You need options in case you get made redundant and London has plenty of options. I realise that this doesn't apply to all careers, but I'm just pointing out that it's a huge reason why people stay here. There are very good salaries to be had in London for those willing to climb and jump around the job market. That combined with a bit of help from family to get started on the property ladder means people can afford to pay £700K+.
Gardens are often less important to people in London because there are many large parks and other attractions for families to engage in at the weekend. Many people prefer to have small courtyard gardens that require less maintenance as they are mainly used for drinking wine in :-D. People are more prepared here to live in houses deemed modest outside of the south east, because more emphasis is put on the cultural and lifestyle benefits of living in such a diverse city. Also, often people don't maintain a car or are happy to drive older cars where they aren't paying out on leases every month. So there are costs to be saved on travel.
I feel like I've almost talked myself out of leaving! I'm headed out to the countryside, because I'm looking for my next chapter. However, I've spent 25 years in this city and if you can make it work for you, it can pay you back a thousandfold.
So as not to ignore the OP. Anywhere on the Windrush Line will get you to Whitechapel for a fast train to Paddington. The Lizzy line has really opened up the travel options. Have a look at Crystal Palace as you will definitely get a 3 or 4 bed house for £900K. The backup route to Paddington from there is train to Victoria and the bus (number 36 I think). When I used to work in Paddington, I actually had a bike stashed in the bike park behind Victoria station (it's in the NCP car park) and I used to cycle the last 3 miles or so to Paddington. So there are multiple options for transport if one line isn't working.
I don't have direct experience of local schools there but friends have raised their kids in the area and they have attended and graduated from the state secondary schools. All appear to be doing well and have very full extra-curricular lives - sports, drama, more niche interests that are well-catered for in London.
Good luck with your search.