Please take another look at the list I gave you in the OP.
equality for slaves
equality for women
equality for homosexuals
equality for the children of the poor
equality for the insane or the handicapped
equality for all minority groups
Legal slavery has been abolished worldwide. The other trends are in the process of spreading from culture to culture. The equality I'm referring to isn't something that I'm saying OUGHT to happen. It HAS BEEN happening. It IS happening. It's a conscience-driven moral upgrade which informs us that all of us are equal in human worth.
You used the rules in competitive sports as an analogy. Why do you think a society, which is a cooperative endeavor, should make survival competitive? Why do you think it fair that someone born with a 150 IQ should be rewarded as though his/her good fortune at birth was an achievement to be rewarded and not a gift to be used in the service of all?
You are confusing equal rights with equal results. All the groups you mention are under the same rules as the person with the IQ of 150. They all have to obey the same speed limit on the road, and they all can get their license at the same age. Some will be better drivers, but nobody is treated differently by the law. Each may behave differently behind the wheel, but this is not due to an outside agent lording over them.
The problem with equal results is it discriminates against someone else, from the outside, since there are now more than one set of rules. This is where injustice occurs. Progressives thought the quota system was just. However, it created more than one set of rules, that targeted even those who never did anything wrong. All this did was swap the injustice in a zero sum game.
In the USA, there are now more females in college and universities than there are men. According to you, this needs to change since this is no longer equal results. We can use quotas, so there are as many men in college as woman. Yet there is no outcry for this. The progressives who preach fair are not concerned with fair, but with an agenda that promotes their voter base. This is where injustice can appear, all dressed up in sheep's clothing.
On the other hand, if we have equal rights=rules for all, and women dominate the college population, then this is natural result, and is not a dual standard agenda from the outside, that violates the rights of others. Men are using more common sense, wanting to avoid the huge debt associated with college education. Now the left wants two sets of rules where the tax payer, who does not go to college, has to pay for the education go those who do not wish to pay their debt. The poor college student needs a free ride at the expense of those who are making ends meet. One that one set of rules is often an agenda driven scam. The Progressive want to play God, rather than accept the God given talents and/or lacks thereof.
In the political system the leaders often get away with crimes and offenses; Russian collusion delusion, since they are above the laws given to everyone else. That is the dark side of more than one set of rules. Socialism creates more than one set of rules, where Big Government has all the power and free enterprise is inhibited.
The Royal family used to make all the rules, but in a way that gave them all the cards regardless of their talent and humanity. One set of rules was an upgrade allowing natural talent to be nurtured, everywhere. This is how nature works. More than one set of rules is regressive and immortal. If is often disguised in sheep's clothing.
The better way to deal with the "disadvantaged" is through charity. Charity can produce the same results while retaining one set of rules for all. You're giving does not take away from another. Progressives do not want to give from their own supplies, but prefer change the rules to they can extort others, and then take credit for caring.
If illegal immigration was important to you, have an illegal immigrant family move in with you. This takes nothing away from anyone else, one set of rules can still apply and the ideal result can occur. It is not about playing god with the rights of others, so you can get credit for caring, without having to give.