Equality of opportunity is, at one and the same time, a worthwhile ideal and an impossible objective. It is a worthwhile ideal to the extent that efforts are made to ensure there are no formal barriers in the way of anyone gaining entry to schools, universities, the various professions and avenues to distinction more generally, and none to their pursuit of success when they've made that entry - no barriers such that some groups may be excluded from any public sphere of activity or have obstacles placed in their way once there. It matters that everyone has the same open field, so to put it, and a society which acts to achieve this as far as possible is better than one that doesn't.
If, however, in such a society there are substantive inequalities of resources - inequalities that stretch between being significant and being huge - then there cannot be genuine equality of opportunity. The many advantages which richer parents can buy for their children, in terms of better physical surroundings, better health care, better education, better everything, are bound to mean that their children have better opportunities than poorer children. It's like a race in which the conditions on the track ensure an equal chance for all participants - same starting point, same finishing line, each person's lane similarly firm and even underfoot - but where all of them have to carry weights, and the weight of those weights varies inversely with the resource-holdings of the runner's family.
I have yet to see a persuasive counter-argument against this standard socialist point on the internal incoherence in the idea of equality of opportunity. Equality of opportunity is, in fact, a misnomer.