The government has very little graft. Politicians seemed to genuinely work for the benefit of their constituents [0]. I always attributed this to the fact that they lived in the communities they served, and the people they socialised with were affected by the decisions they made.
The UK has the problem that the politicians are insulated from the people they ostensibly serve. Especially the Tories. They have displayed contempt and arrogance beyond belief.
This is all, as you say, a problem of scale.
[0] One illustration of this is the Manx implementation of social security vs the UK. When the Manx government set up national insurance contributions, they paid them into a separate fund (https://www.gov.im/about-the-government/departments/the-trea...) which is under control of the treasury. The UK just poured the national insurance contributions into the general taxation pool.
The Manx fund (and by extension their health service) is rolling in money. The UK NHS is broke. The Manx system is proof that there were more than enough contributions to the NHS to keep it adequately funded. But the UK politicians spent the contributions on other things, because they could.