I think that looking at this is the terms of a 'lottery' is nonsense. Winning a lottery is pure luck. It doesn't depend on any way on the effort put in by the winner. All they have to do is enter the lottery.
Children aren't voluntary entrants to this 'lottery' ( if we have to frame life chances in this way) They are born into it and indeed where they end up is a matter of sheer luck. We should be able to do better than just shrug our shoulders and say that 'that's life'.
We should at least be able to try to ensure that, to use another metaphor, they all start on a level playing field and level playing fields are dependent on the effort put into them to level them and keep them level.
Someone said earlier that we have the NHS and free education, what more can governments do? But both of these contain large bumps which prevent people achieving good health care and education; those bumps being the result of government action in withholding the funds needed to run both effectively and accessible to all.
We have to make up our minds. Do we want a state which cares enough for its citizens to focus its resources and its spending power on promoting health and a decent education (and other vital services such as justice and observation of the rule of law) for all or do we want one where the weak and disadvantaged are left to sink?
Anyone who has studied social history will know that we suffered hundreds of years of the latter and we enjoyed an all too brief 'blip' in this reality in the post WW2 period when then 'state' was determined to change the status quo. Implementing as best it could the Beveridge report's attempt to deal with the 'giant evils' of want, disease, ignorance, squalor, and idleness. I know this is boring repetition, but Thatcher put a stop to that as fast as she could and we struggle on with the results...