Did the United States live up to its founding ideals of liberty, equality, and justice in the period from the Gilded Age to the Age of Obama?

Largely yes, but with some flaws as well.

Explanation:

The United States is nowadays a developed, high-income country, where the vast majority of citizens have a quality of life way higher than the world average.

The country also has a political system that works better than most countries in the world. The judicial branch is largely independent from the president, and this is not something to take as a given.

The country also has a great degree of liberty in many areas: religion, ideology, political association, and so on, which again, is also higher than the average country in the world.

However, there are still flaws, especially when it comes to equality. In the United States, there is great income inequality, and this inequality has been rising in the last decades. Racial minorities are also subject to some other forms of inequality that are more institutional, which makes it harder for them to progress.

Overall, the country mostly lives up to its ideals when compared to the vast majority of countries in the world, but it still has its flaws, and should improve.