The government DOES, however, guarantee equality of opportunity

I thought you said the only equality that matters is equality before the law. That’s a bit different than equality of opportunity.

Embracing inequality is when those who don't get something are told that's the way it always was, always will be, is the natural order of things, they're forbidden from it or get it second or third string, and that's why society is great

I don’t mean to parse your words too fine, but do you mean to say that equality depends on people getting things, while inequality depends on people being told they shouldn’t get things?

Embracing equality depends on what society is trying to rectify.

This implies an injustice necessarily accompanies inequality. Is this what you intend?

It can be as simple as getting to vote when they previously could not.

Equality of political rights.

In more extreme cases it can be wealth and property reassignment like what happened after Apartheid ended.

Equality of a particular resource distribution.

This response taken as a whole tells me you favor equality in all its various forms, but equality before the law (vote), or equality of ownership (land redistribution). If this is incorrect please clarify.

The equality I mean is one whereby we are all equally working for the common good and benefiting equally from society's wealth. Everyone's work is equally valued and everyone's need is equally met.

I put this quote first because I will assume this is the equality you meant throughout your post.

Your view is unjust. The type of equality you propose would necessitate the removal of Liberty, which is a more important core value that makes society fair and just.

Work is too dynamic to say that everyone could be equally working. Is an authors work equal to that of a garbage mans? No. Furthermore, this kind of equality would require that people be compelled to work lest they gain their share of societal wealth without putting in their equal share of effort. That’s called tyranny of the majority and is slavery.

Everyone’s work cannot be valued equally for a similar reason that the amount of work cannot be compelled to equality. Not all work is as valuable as all other work. A burger flipper could put in way more hours than an engineer and still not produce the kind of value an engineer will. That’s because the burger flippers work is not as valuable. It can’t be made to be so.

People cannot be made to work equally. People cannot be made to value work equally. Society cannot provide for needs properly, let alone equally, if work and value is equalized. This is because people will minimize their production when they know they will see an equal return regardless of effort. This isn’t my pet opinion, we observed this in literally every socialist experiment.

Equality is a virtue because it is a core value which makes society fair and just

If you said equality before the law, I would have agreed. People do not possess the same talents, drives, origins, or values. To equalize people in the way you propose, is to negate the humanity of each individual. Which would be fine since those individuals success doesn’t matter. . Your idea of equality is the opposite of fair and just.

Put a hero next to a villain and call them equal. Who benefits?

Put shoddy construction worker next to a master craftsman and call them equal. Who benefits?

Put poison next to your food and call it equal. What happens?

when there is a sense that we are all in it together we can work for the common good

People already do feel that they are in it together with other members of their chosen group. They often work as teams to find group success, and equality within the group is not essential. It’s often a hindrance. The reason people do this, even when there is inequality in the group, is because those in the group individually value the common Good to be achieved. Creating equality of persons, means, and ends does nothing to unite humans, but rather it works to dehumanize slaves.