The impact of family income largely depends on what part of the educational cycle is being considered. This really only applies to secondary level education, where there is a proven link between income and attainment. However family income is only a suitable measure, it hides a far more complex relationship.
For tertiary education a loan system is open to all and repayment rates are coupled to subsequent income level. So family income should have a negligible impact for most.
For post graduate degrees the availability of funding largely depends on the subject matter. The more industry orientated the degree, the greater availability of funding
If you are at the bottom of the heap on a sink estate you face some severe challenges trying   to obtain a good education. Breaking that relationship requires political will and serious investment. Unfortunately life is not fair and never has been. Lip service has been paid to keeping such kids healthy, fed and clothed but little has been done to offer opportunity and hope for them. There is a token acceptance of their rights, but nothing concrete to guarantee them.
We abuse pensioners in care   and deny basic rights to many impoverished children. So much for a caring society!