Helping to buy - what? A shed?

The government has a few schemes that supposedly assist first-time house buyers, including help-to-buy loans of 20 or 40% of the value of a home.
These are a bit more generous in London: help-to-buy loans are available on properties worth up to £450k instead of £250k; and the starter homes and shared ownership schemes require a household income of less than £90,000 rather than £80,000.
The average house price in London is £474k, or 105% of the scheme maximum.
The average house price in the rest of the country is £210k, or 84% of the scheme maximum.
No surprises then that buyers in London don't feel very helped. Why is the London weighting not weighted for London? What's the point?

House location in other parts of the country matters less. Thinking of the cities I know, if I needed to live a little further out in order to be able to afford a home, it would change a commute into the city centre from a 10 to a 15 minute cycle ride; hardly life-changing. In London being pushed further from where prices are 120% of average to 95% will take someone possibly 20-30 minutes further out of the city, which is itself big enough to need a 30-minute commute if you're not lucky enough to have a job on the edge of the city closest to you.
An extra 30 minutes on top of whatever commute you already had will change a life. A decent cycle ride turns into a training regime for elite competition that gets you home late and filthy - and the same into work. A hope on the Tube turns into reliance on Southern Rail, infamous for its customer service. Going out in the evening can no longer be a quick hope from home; it needs to be planned so that you can go straight from work, because if you pop home you'll never get anywhere. Anyone who might wear different clothes (or hair, or make-up) from at work will need to carry stuff around and prepare themselves in the work loos. And once out the evening is limited by the last train, because taxis, expensive anyway, are prohibitively so when travelling miles in the middle of the night.
Or, if you don't care about the intangible effects on life, we can find a lower bound on the value by reference to salaries. It's another hour a day doing nothing fun. The average London salary is about £37k p.a., which is about £18 per hour. That's a mere £4,000 a year that living further out costs (on top of the season tickets etc.).
This is what help-to-buy is, even ignoring the (justified) criticisms that it has put young people in greater debt in order to provide profit to housebuilders. It's a scheme that encourages young people out into suburban ghettoes, asking them to live the commuting lifestyle of middle-aged people in areas determined by price not community. And then people find it strange to see statistics showing that young people go out and enjoy themselves far less than previous generations, or that people feel less and less sense of community.
In short, 'London weighting' shouldn't just be a finger-in-the-air guess (or head-up-your-own-rear guess). We have national statistics! Government policy should probably use them. Maybe someone did, but incompetence seems more likely than an even more complex assessment.