I would say you're right to question this assumption. When incomes rise I'd expect demand for government services such as health and education to lower as individuals sourced healthcare and their child's education from the private sector. As government essentially provide services for 'free' anyway (i.e you're not paying for schooling/health at the point of use) I don't see why changes in income would have that much of an impact. A rise in my monthly pay packet makes absolutely no difference to my consumption of the free local newspaper that drops through my letterbox everyweek, the same is surely true with government services.
The general assumption is that as income for people increases, the GDP in the country increases, so growth will increase and government will benefit from receiving more money ie from taxes etc...so eventually they will use that money to fund public services
Perhaps poorer people have imperfect knowledge due to market failures and thus demand less government based services? Many merit goods are underconsumed as the poor do not realise the true value of it's consequences.