People most definitely DO NOT have similar values, and this almost entirely explains the left/right divide within the United States. Most Americans have been brought up to believe in shared beliefs like individual liberty, equality before the law, basic human rights, etc. This is the common vision we are striving to achieve. But beliefs are not the same as values. The divide comes in how we prioritize our values in achieving those goals.

Jonathan Haidt has done a lot of important work in this area, I recommend you check him out. Our moral foundations are based on: Harm/care, Fairness/reciprocity, Ingroup/loyalty, Authority/respect, and Purity/sanctity. Lefties really only care about the first two, conservatives care about all 5, and harm/care + fairness/reciprocity typically rank lower than the other values. Look at the FIRST comment on this thread. Then look at the list of values. It is really easy to see what our arguments are about once you know why people think the way they do.

I disagree strongly with your last point. Almost everyone wants to be happy, avoid suffering, have healthy and harmonious relationships, to live life as they choose, have a sense of purpose, etc. Our fights are typically about how to achieve that for every citizen. If you can find a LARGE population of people that does not want those things I will happily retract my statement.