As a fairly in-depth starting point, the modern Oxford Histories do a very good job for giving a balanced approach. They are the new and updated Oxford Histories from the originals written in the 1960s, and the series is still being written I believe.

The Glorious Cause, Robert Middlekauff (1763-1789)

Empire of Liberty, Gordon S Wood (1789-1815)

What Hath God Wrought, Daniel Walker Howe (1815-1848)

Battle Cry For Freedom, James M McPherson (1848-1865)

The Republic for Which It Stands, Richard White (1865-1896)

Freedom From Fear, David M Kennedy (1929-1945)

As I say, others in the series are being written which explains the gap, and there are books for the later C20th as well.

What I would say is that you will never find a history book that is not ‘biased’ to some degree. We are all biased by our upbringing, where we are born and live, our skin colour or gender etc, but trained and professional historians can hopefully recognise these biases and be conscious of them.

One book that may not fit that ‘unbiased’ criteria that you speak of, but I will always recommend is Howard Zinn’s ‘A People’s History of the United States’. It is a general chronological covering of the the modern day US from the 1490s to the end of the C20th, and it is an excellent read.

Similarly, for the Cold War era, you cannot go wrong with John Lewis Gaddis’ ‘The Cold War’, which obviously touches on Asian and European history as well, but gives a very strong American focus on the post-war settlement.

I hope this is helpful!

Edit: formatting went weird.