I think that Jefferson, who was the main writer of the US Constitution, meant what he wrote and that he meant happiness as we understand it.
He did mean what he wrote, but he didn’t mean “happiness” as we understand it, in the sense of a feeling of pleasure or contentedness. That’s a fairly modern sense of the word.
“Happy” is related to the word “happen”, and in its original sense it referred to something which occurred by chance or by fortune – not necessarily a good thing. A “happy event” originally did not mean an event that caused rejoicing, but rather an event that need not have occurred at all.
From there the meaning shifted slightly. First, it began to refer to events which happened not simply by fortune, but by good
fortune. Secondly, the word was used to refer not only to fortunate or promising events, but to the person to whom fortunate or promising events happen, or are going to happen. It’s used in this sense in the King James Bible; “happy are the poor in spirit” and “happy are those who mourn” doesn’t mean that they have a feeling of pleasure and contentment; they quite clearly don’t. It means that good things are going to happen for them.
From there we get the third shift of meaning, to refer to the pleased or contented state of mind that we experience when good things happen to us. This meaning had developed by Jefferson’s time, but it hadn’t then become the dominant meaning of the word.
I don’t think Jefferson would have understood the Declaration of Independence to mean that we have a natural right to feel pleased and contented, in parallel with our natural right not to be locked up, and our natural right not to be shot. Whatever his views on the role of the state in relation to property, he certainly didn’t think it was the role of the state to induce a feeling of mental well-being. Apart from anything else, in the Age of Enlightentment and Rationality which gave rise to the American revolution, feelings, moods and states of mind were simply not that important; it was important that men should do good, not that they should feel good. It’s in the later Romantic era that attention is paid to sentiments.
We also have to think about what it was the American revolutionaries were rebelling against. The British government of the colonies was not attempting to control anybody’s state of mind or to try to dictate to anyone what the good things in life were or to stop anybody from acting virtuously. But they were attempting to control trade and economic activities through taxation, through the grant of monopolies, through preventing colonial businesses from competing with home businesses, etc. The colonists resented this because they felt that it stopped them from pursuing their own well-being, in order to protect the entrenched advantages of others.
So I’m still of the opinion that the right to “the pursuit of happiness” was essentially the right to take advantage of opportunities that presented themselves, and these were mostly conceived of in terms of economic opportunities.