The moral and practical thing to do would be focusing on improving objective outcomes. Focusing on equality degrades objective outcomes in favor of moral perversion - taking away from some to make others feel better.
Imagine the child of two millionaire college professors. What will you do to equalize this child's opportunity with a child in the inner city of (pick a bad city)?
Improving objective outcomes is some Krypton level shit, "Your a black doctor and were trained to be one from birth due to the outcome we wanted." "I wanted to be an artist."
You meant at t0, the movement the baby is born to a parent optimizes education vs a baby that is born to a parent that cares much less. Impossible for government to "make sure" this happens.
There is no real way to perfect this - but if an effort is made a good result can be achieved nonetheless. The fallacy with the professor example is that they will have the time to teach their child, and that their child must necessarily be receptive to their limited tuition.
Unless they quit their job, and thus are no longer professors, a majority of their time will be spent teaching multiple students, and while one on one coaching has it's merits there is also value in classroom peer experiences. If an effort is made to provide tutoring to all the kids, it may even be in the parents interest not to provide tutoring services, if the quality is good, because that takes away from their own time spent being able to provide the most effective teaching experience they can offer.
We can't build perfect bridges, but we can sure as hell mandate they are good enough, it's some nonsense to say we shouldn't try because we can't get it perfect.
Or do you simply deny that parents educate their children? Do you believe that as long as they have the same teachers at school, who are 'paid properly', say, a farmer's son and a son of a musician will have the same knowledge and skills in animal husbandry and musical theory?
That is obvious, the discussion is about the education opportunities though, not mastering school math, which is, honestly a joke.
Equality of outcome can be an undesired outcome, equality of opportunity is completely different.
The only trouble is getting the opportunities to be equal - there must be no advantage that can be unfairly given to one more deserving student than another, being able or willing to send your children to summer bootcamps must be an option for all children (who qualify), not whether you pay for it with time or money, e.g.
Free college is fantastic, but once again you confuse opportunity with outcome, and not even for the same individuals - parents are unburdened by cost, but in fact the opportunities are far from equal - money does not a quality education make, yet the majority or colleges are run as for profit institutions, not places that accept students based on their merits or potentials, nor do they actually try to actively shed students who are undeserving. Party culture does not need or require an expensive room and board situation, yet it pervades nearly every 'higher' education institution, only somewhat subsiding when graduate/doctorate programs become involved, and academics are once again taken seriously.
I.e. your meritocracy does not exist precisely because universities are busy making profits not teaching students.