I think your logic here is flawed. Just because a few people abuse the concept of human rights doesn't make the concept itself false.
As Christians we understand that everything we have is given to us by GOD. From the breath in our lungs to the food in our bellies, even our life itself.
So, GOD grants life. No one claiming to be a Christian can deny that.
Liberty. A person's natural desire is for freedom. Of course that desire gets corrupted because we are sinful, so liberty in this sense is NOT the freedom to pursue immoral or evil desires. It's the freedom to pursue the desires that GOD has placed in each and every one of our hearts. This is something that is a bit more difficult for non-Christians to understand, but it should be another no-brainer for Christians.
This is also where the two greatest commandments come into play: love GOD above all else and treat others the way you would like to be treated. If you keep those two commandments, then you can never abuse your liberty. The Founding Fathers understood this and it's why they warned us that the great American experiment would fail if our country abandoned the Christian values that we were founded upon.
The pursuit of happiness is simply the drive to live our lives according to the unique design that GOD has given each and every one of us. For some that might be raising a family, for others it might be art or music, still others might have a bent towards animal husbandry or farming. The list goes on and people can even have combinations of multiple GOD-given desires.
Governments can only infringe upon those basic rights and they do that with rigor, all the time. Even America itself has become corrupted to the point where American citizens don't even understand what this country was founded upon. Some of those people end up in the government and they might even end up in the highest seats of government (like the Presidency for example).
So, it's an ideal that America was founded upon; but that doesn't mean that we are always the best representatives of that ideal.